Once on each of your turns as a bonus action, you can mentally
command the servant to move up to 15 feet and interact with an object.
The servant can perform simple tasks that a human servant could do,
such as fetching things, cleaning, mending, folding clothes, lighting
fires, serving food, and pouring wine. Once you give the command, the
servant performs the task to the best of its ability until it
completes the task, then waits for your next command.
(PHB p. 284)
What happens if the Unseen Servant e.g. tries to steal gold out of a pocket?
Or if he is commanded to pick a lock? I would consider those as "simple tasks that a human servant could do".
Can an Unseen Servant do actions that require ability checks (apart from attacking, which is explicitely prohibited) and if yes, does the caster add his relevant modifier and proficiency bonus?
Best Answer
Simple Tasks are not Ability Checks
You may come to a table decision allowing for an Unseen Servant to perform more than just the "simple tasks" listed, but the assumption is that a simple task is not something that requires an ability check.
Compare this to the Arcane Trickster's Mage Hand Legerdemain (PHB, 98):
Legerdemain specifically allows for certain ability checks while Unseen Servant is merely for simple tasks.