Setting JLabel text to be random string from an ArrayList

Question 1

You have a reference problem...

In your main method, you declare three local variables;

public static void main(String[] args) {
    ArrayList<String> SPOquestions = new ArrayList<String>(); // adding the sports questions for the quiz
    //...
    ArrayList<String> MUSquestions = new ArrayList<String>(); // adding the music questions
    //...
    ArrayList<String> GENquestions = new ArrayList<String>(); // adding general knowledge questions
    //...
}

This means that this variables can not be referenced from any other context other than main. A "simple" solution might be to make them static class variables of the Questions class. While this would work, this creates problems and represents a larger, long term issue, static is not answer to allowing access to variables, but represents a design problem.

Instead, move the variables to the class that needs to use them and create them as instance variables, so that they are unique to each instance of the class...

public class QuestionPane extends JPanel {

    private ArrayList<String> SPOquestions = new ArrayList<String>(); // adding the sports questions for the quiz
    private ArrayList<String> MUSquestions = new ArrayList<String>(); // adding the music questions
    private ArrayList<String> GENquestions = new ArrayList<String>(); // adding general knowledge questions

In this way, QuestionPane can access each of the variables from within it's context (any method declared within QuestionPane)

For example...

import java.awt.BorderLayout;
import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.GridBagConstraints;
import java.awt.GridBagLayout;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.util.ArrayList;
import java.util.Collections;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JLabel;
import javax.swing.JPanel;
import javax.swing.UIManager;
import javax.swing.UnsupportedLookAndFeelException;

public class RandomStuff {

    public static void main(String[] args) {
        new RandomStuff();
    }

    public RandomStuff() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                try {
                    UIManager.setLookAndFeel(UIManager.getSystemLookAndFeelClassName());
                } catch (ClassNotFoundException | InstantiationException | IllegalAccessException | UnsupportedLookAndFeelException ex) {
                }

                JFrame frame = new JFrame("Testing");
                frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
                frame.setLayout(new BorderLayout());
                frame.add(new QuestionPane());
                frame.pack();
                frame.setLocationRelativeTo(null);
                frame.setVisible(true);
            }
        });
    }

    public class QuestionPane extends JPanel {

        private ArrayList<String> SPOquestions = new ArrayList<String>(); // adding the sports questions for the quiz
        private ArrayList<String> MUSquestions = new ArrayList<String>(); // adding the music questions
        private ArrayList<String> GENquestions = new ArrayList<String>(); // adding general knowledge questions

        private JLabel questionLabel;
        private JButton randomButton;

        public QuestionPane() {

            setLayout(new GridBagLayout());
            GridBagConstraints gbc = new GridBagConstraints();
            gbc.gridwidth = GridBagConstraints.REMAINDER;

            questionLabel = new JLabel();
            randomButton = new JButton("Next question");
            randomButton.addActionListener(new ActionListener() {
                @Override
                public void actionPerformed(ActionEvent e) {
                    String text = "?";
                    switch ((int)Math.round(Math.random() * 2)) {
                        case 0:
                            Collections.shuffle(SPOquestions);
                            text = SPOquestions.get(0);
                            break;
                        case 1:
                            Collections.shuffle(MUSquestions);
                            text = MUSquestions.get(0);
                            break;
                        case 2:
                            Collections.shuffle(GENquestions);
                            text = GENquestions.get(0);
                            break;
                    }
                    questionLabel.setText(text);
                }
            });

            add(questionLabel, gbc);
            add(randomButton, gbc);

            SPOquestions.add("Who won the 2005 Formula One World Championship?");
            SPOquestions.add("Which team has the most Formula One Constructors titles?");
            SPOquestions.add("In what year did Roger Federer win his first 'Grand Slam'?");
            SPOquestions.add("How many 'Grand Slams' has Rafael Nadal won?");
            SPOquestions.add("Who has scored the most amount of goals in the Premier League?");
            SPOquestions.add("Who has won the most World Cups in football?");
            SPOquestions.add("How many MotoGP titles does Valentino Rossi hold?");
            SPOquestions.add("Who was the 2013 MotoGP champion?");
            SPOquestions.add("Who won the 2003 Rugby World Cup?");
            SPOquestions.add("In rugby league, how many points are awarded for a try?");
            SPOquestions.add("Who is the youngest ever snooker World Champion?");
            SPOquestions.add("In snooker, what is the highest maximum possible break?");
            SPOquestions.add("How many majors has Tiger Woods won?");
            SPOquestions.add("In golf, what is the tournament between the USA and Europe called?");
            SPOquestions.add("How many World Championships has darts player Phil Taylor won?");
            SPOquestions.add("What is the maximum possible amount of points a player can earn from throwing three darts?");
            SPOquestions.add("How many gold medals did Michael Phelps win at the 2008 Beijing Olympics?");
            SPOquestions.add("Who won the 2012 Olympic 100 metres mens race?");
            SPOquestions.add("Which of these events are not a part of the heptathlon?");
            SPOquestions.add("When was the first modern Olympics held?");

            MUSquestions.add("'Slash' was a member of which US rock band?");
            MUSquestions.add("Brian May was a member of which English rock band?");
            MUSquestions.add("What is the name of the music festival held annually in the town of Boom, Belgium?");
            MUSquestions.add("The rapper Tupac Shakuer '2Pac' died in which year?");
            MUSquestions.add("Which of these bands headlined the 2013 Glastonbury music festival?");
            MUSquestions.add("Which of these people designed the 'Les Paul' series of guitars?");
            MUSquestions.add("Keith Moon was a drummer for which English rock band?");
            MUSquestions.add("Kanye West has a total of how many Grammy awards?");
            MUSquestions.add("Beyonce Knowles was formally a member of which US group?");
            MUSquestions.add("In which US city was rapper 'Biggie Smalls' born?");
            MUSquestions.add("Michael Jackson's first number one single in the UK as a solo artist was what?");
            MUSquestions.add("The best selling album of all time in the UK is what?");
            MUSquestions.add("The best selling album of all time in the US is what?");
            MUSquestions.add("What is the artist known as 'Tiesto's real name?");
            MUSquestions.add("Which of these was not a member of The Beatles?");

            GENquestions.add("Who was the second President of the United States?");
            GENquestions.add("The youngest son of Bob Marley was who?");
            GENquestions.add("In the film '8 Mile', the character portrayed by Eminem is known as what?");
            GENquestions.add("What is the capital city of New Zealand?");
            GENquestions.add("What is the capital city of Australia?");
            GENquestions.add("How many millilitres are there in an English pint?");
            GENquestions.add("What was the biggest selling game for the PS2 worldwide?");
            GENquestions.add("What is the last letter of the Greek alphabet?");
            GENquestions.add("Who created the television series Futurama?");
            GENquestions.add("A word which reads the same backwards as it does forwards is known as a what?");
            GENquestions.add("A 'baker's dozen' consists of how many items?");
            GENquestions.add("World War 1 officially occured on which date?");
            GENquestions.add("'Trouble and strife' is cockney rhyming slang for what?");
            GENquestions.add("Who was the last Prime Minister to hail from the labour party in the UK?");
            GENquestions.add("WalMart is the parent company of which UK based supermarket chain?");

            randomButton.doClick();
        }

        @Override
        public Dimension getPreferredSize() {
            return new Dimension(400, 200);
        }

    }

}

You may also want to take a look at Code Conventions for the Java Programming Language, it makes it easier for others to read you code ;)

You may also want to have a read through Language Basics: Variables, Understanding Class Members and Creating a GUI With JFC/Swing

Question 2

Try using

main.MUSquestions

instead of

MUSquestions

Or replace main with what ever your Class you created the List in is named.

Or, to keep the references shorter and make it faster, declare a new list by adding

ArrayList<String> MUSquestions = main.MUSquestions

to what is in your question the upper Class - like what you have written down it can't find MUSquestions because it is not defined as it is only available in another class. (like @MadProgrammer said)

Question 3

Here I've created a new answer:

This is a main class to launch the application, and to show usage of the Questions class I have made.

import javax.swing.JLabel;

public class Main {

    Questions questions = new Questions();

    public static void main(String[] args) {
        new Main();
    }

    public Main() {
        JLabel label = new JLabel();
        label.setText(questions.getRandomQuestion(questions.getSpoQuestions()));
    }
}

This is the Questions class who provides some methods and arraylists with their questions.

import java.util.ArrayList;
import java.util.Random;

public class Questions {

    private ArrayList<String> spoQuestions = new ArrayList();
    private ArrayList<String> musQuestions = new ArrayList();
    private ArrayList<String> genQuestions = new ArrayList();

    public Questions() {
        createSpoQuestions();
        createMusQuestions();
        createGenQuestions();
    }

    public ArrayList<String> getSpoQuestions() {
        return spoQuestions;
    }

    public ArrayList<String> getMusQuestions() {
        return musQuestions;
    }

    public ArrayList<String> getGenQuestions() {
        return genQuestions;
    }

    public String getRandomQuestion(ArrayList<String> list) {
        return list.get(new Random().nextInt(list.size()));
    }

    private void createSpoQuestions() {
        spoQuestions.add("Who won the 2005 Formula One World Championship?");
        spoQuestions.add("Which team has the most Formula One Constructors titles?");
        spoQuestions.add("In what year did Roger Federer win his first 'Grand Slam'?");
        spoQuestions.add("How many 'Grand Slams' has Rafael Nadal won?");
        spoQuestions.add("Who has scored the most amount of goals in the Premier League?");
        spoQuestions.add("Who has won the most World Cups in football?");
        spoQuestions.add("How many MotoGP titles does Valentino Rossi hold?");
        spoQuestions.add("Who was the 2013 MotoGP champion?");
        spoQuestions.add("Who won the 2003 Rugby World Cup?");
        spoQuestions.add("In rugby league, how many points are awarded for a try?");
        spoQuestions.add("Who is the youngest ever snooker World Champion?");
        spoQuestions.add("In snooker, what is the highest maximum possible break?");
        spoQuestions.add("How many majors has Tiger Woods won?");
        spoQuestions.add("In golf, what is the tournament between the USA and Europe called?");
        spoQuestions.add("How many World Championships has darts player Phil Taylor won?");
        spoQuestions.add("What is the maximum possible amount of points a player can earn from throwing three darts?");
        spoQuestions.add("How many gold medals did Michael Phelps win at the 2008 Beijing Olympics?");
        spoQuestions.add("Who won the 2012 Olympic 100 metres mens race?");
        spoQuestions.add("Which of these events are not a part of the heptathlon?");
        spoQuestions.add("When was the first modern Olympics held?");
    }

    private void createMusQuestions() {
        musQuestions.add("'Slash' was a member of which US rock band?");
        musQuestions.add("Brian May was a member of which English rock band?");
        musQuestions.add("What is the name of the music festival held annually in the town of Boom, Belgium?");
        musQuestions.add("The rapper Tupac Shakuer '2Pac' died in which year?");
        musQuestions.add("Which of these bands headlined the 2013 Glastonbury music festival?");
        musQuestions.add("Which of these people designed the 'Les Paul' series of guitars?");
        musQuestions.add("Keith Moon was a drummer for which English rock band?");
        musQuestions.add("Kanye West has a total of how many Grammy awards?");
        musQuestions.add("Beyonce Knowles was formally a member of which US group?");
        musQuestions.add("In which US city was rapper 'Biggie Smalls' born?");
        musQuestions.add("Michael Jackson's first number one single in the UK as a solo artist was what?");
        musQuestions.add("The best selling album of all time in the UK is what?");
        musQuestions.add("The best selling album of all time in the US is what?");
        musQuestions.add("What is the artist known as 'Tiesto's real name?");
        musQuestions.add("Which of these was not a member of The Beatles?");
    }

    private void createGenQuestions() {
        genQuestions.add("Who was the second President of the United States?");
        genQuestions.add("The youngest son of Bob Marley was who?");
        genQuestions.add("In the film '8 Mile', the character portrayed by Eminem is known as what?");
        genQuestions.add("What is the capital city of New Zealand?");
        genQuestions.add("What is the capital city of Australia?");
        genQuestions.add("How many millilitres are there in an English pint?");
        genQuestions.add("What was the biggest selling game for the PS2 worldwide?");
        genQuestions.add("What is the last letter of the Greek alphabet?");
        genQuestions.add("Who created the television series Futurama?");
        genQuestions.add("A word which reads the same backwards as it does forwards is known as a what?");
        genQuestions.add("A 'baker's dozen' consists of how many items?");
        genQuestions.add("World War 1 officially occured on which date?");
        genQuestions.add("'Trouble and strife' is cockney rhyming slang for what?");
        genQuestions.add("Who was the last Prime Minister to hail from the labour party in the UK?");
        genQuestions.add("WalMart is the parent company of which UK based supermarket chain?");
    }
}

These two pieces of code are an entirely working application. I didn't make a GUI context as the purpose is to show you how to use it in a label.

The method

    public String getRandomQuestion(ArrayList<String> list) {
        return list.get(new Random().nextInt(list.size()));
    }

will return a String, randomly pulled from the list given with the method. So, looking at the Main class example, you can see that I pull a random question from the spoQuestions ArrayList by calling the getter for that list.