Been there, done that...
In the last house, I suffered (a) regular mole incursions to my lawn (b) do-gooder neighbours, who had opinions that were not based on actually having to deal with the moles.
I tried all sorts of stuff including various "humane" solutions, most of which caused some moles to die of laughter, and the rest to move in on me.
What worked in the end: have you discovered rook scarers? Out in the countryside, us locals can still buy serious bangers intended to emulate a 12-bore shotgun sound. (You poor nannied townies of course can't be trusted with toy firecrackers on fireworks night.)
Find a mole run (a series of mole hills in a more-or-less straight line). Dig a mole trap (scissors-type preferably) into the middle one, observing precautions against tainting with human smell. Then dig into both end hills, put a rook scarer into the tunnel at each end, fire them off (cover your ears if you want to be able to hear afterwards). This deafens the mole (who then can't find food) - he then blunders about and stumbles into the trap and gets killed.
Result: 6 dead moles in first week of this method, and a number of complaints from do-gooder neighbours brushed off.