The Trent powered 777/787s have an engine surge EICAS message as an aid to diagnosis.
Not on the ones I fly (777 Trent 895); there may be on the 787? RR engines often have automatic relight/surge control.
Both the Eng Lim/Surge/Stall and Eng Svr Damage/Sep checklists on the 777 are unannunciated, which means you have to do the diagnosis and selection yourself.
In terms of severe damage, with most engines I’d be looking for things like no rotation in any of the spools, bad vibration, missing engine parameters, etc. Loud noises, flames and the like can be indicative of surge/stall but to get to that point the engine has often suffered a fair amount of damage. The Lim/Surge/Stall checklist will lead to shutting it down if the conditions persist and if in doubt you can always call for the Svr Damage/Sep checks...