When the property is used concurrently in a multi-user or multi-threaded environment, could it cause problems?
Absolutely. List<T>
is not designed for multiple threads, except for the case where there are just multiple readers (no writers).
Do I need to use thread synchronization for the code within the property 'LogLevels'?
Well that's one approach. Or just initialize it on type initialization, and then return a read-only wrapper around it. (You really don't want multiple threads modifying it.)
Note that in general, doing significant amounts of work in a static constructor is a bad idea. Are you happy enough that if this fails, every access to this property will fail, forever?