Yes. I have installed the community edition on Tomcat 7. I am able to login but I still testing it. Try this tutorial: http://pentaho-bi-suite.blogspot.com.es/2014/01/installation-of-pentaho-501-ce-with.html
I also took a lot of ideas from here:
http://pablotips.blogspot.com.es/p/installing-tomcat-and-pentaho.html
Some tips:
The Tomcat distribution come with some folders. Copy the Data, pentaho-solutions and licenses to the same directory as tomcat. You will need it.
Do not change the name of the the Tomcat they give you: C:\Pentaho\server\biserver-ee\tomcat\webapps but if you change it you need to update solution-path in web.xml
You need to start the Tomcat with enough memory. Copy the parameters from the start-pentaho.bat. Update the service or the start script.
You need to configure the "_PENTAHO_JAVA" and "_PENTAHO_JAVA_HOME" variables
In tomcat\webapps\pentaho\META-INF\context.xml docbase should be docBase. This will avoid one warning but I still find errors and warnings in the logs.
In \webapps\pentaho\docs there is a lot of documentation that can be helpfull.
Before point 4 "Executing MySQL .sql scripting files for Pentaho Integration" the server should be able to start and work with some demo data in hsql.
On point 5.III of the tutorial it tells who to change the META-INF/context.xml highlighting the change of URL but forgot to highligth the change of driver to MySQL and validationQuery.
If you find some Quartz initialization problems see create an empty table
Good look and please report any other finding.