As per the Jetbrains Resharper documentation on 'Searching a Code with Pattern':
Pay attention, that when you use a placeholder, its name should be enclosed with dollar signs (use the syntax $xx$, where xx represents placeholder name),whereas when you create a new placeholder, you should omit these special symbols.
Therefore, I was on the right track. Also for the placeholder I just need '$anyString$' and it will find all invocations of the method, even if they are made up multiple string objects (e.g. string literals and string objects). So this is how it would look:
The 'anyString' placeholder pattern was created by performing the following steps:
1) Click 'Add Placeholder' -> Argument
2) Give it a name, e.g. 'anyString'
For my case, I also checked the 'Limit minimal number of arguments' and selected 1, and I also checked the 'Maximal' box and set that to 1 also.
The 'Save' button is also useful if you intend on reusing the pattern again.