(Thanks for the ask @mrblade65654654!)
Scout: Telling that one joke to Pauling, it may have canceled his chances at a date
‘Urgh, why did I bring up that lemur and Montana?!’
Engineer: He has 11 PhDs. ONLY 11! Urgh, how embarrassing…
Sniper: Telling his parents he was working as a mercenary… Dark day for Mr. Mundy…
Soldier: HE ONLY PUNCHED 327 NAZIS IN HIS TOUR IN POLAND
or at least who seemed like a Nazi, his helmet was too low to see.. That’s just embarrassing.
Demoman: His mum never lets him forget the fact he still has an eye. Jeez, mum, HE KNOWS!
Medic: Well, I can tell you what he DOESN’T regret… Losing his Medical Licence. Worth! IT!
Pyro: Never having enough lollipops for all the babies on the battlefield, they always feel so guilty that not everyone got one :( They hope the bubbles were good enough though!!!
Heavy: Leaving his family behind in Russia, though he’s glad they don’t have to see what kind of work he does.
Spy: (I actually can’t think of one for Spy, he’s very good at either hiding his emotions or faking them… So, by canon standards, there is no concrete evidence of Spy actually showing regret over anything…
This including the Dad situation.)