I think your question was downvoted for lack of research but I can see how it could be confusing for a newcomer. FullCalendar doesn't need to "support LWC". However, it is a 3rd party library you can use.

There's more to it than a snippet of code as you would upload a static resource, etc but there are projects you can look through or deploy to your dev org to get familiar with how it works.

Salesforce documentation on loading 3rd party libraries:

https://developer.salesforce.com/docs/component-library/documentation/en/lwc/lwc.js_third_party_library

Projects using FullCalendar:

https://github.com/effordDev/sf-calendar

https://github.com/markslott/lwc-fullcalendar

So to answer the question, an exhaustive list of every version of every library that could be used is unrealistic so if you actually need a particular version of any library for some reason and a quick search doesn't tell you then you'll have to just test it. Good news is once you understand loading libraries, it doesn't take long. *Also, yes it's free.

Hope it helps.