Do modern subs ever come to rest on the sea floor?

[Kapitan]

Can modern subs come to reset on the bottom of sea floor?

some nuclear boats will have water intakes on the bottom so sitting on the bottom would clog them up.
That and they don't truly know whats down there below the keel, it may be muddy or rocky either way you don't want to sit a $1bn worth of kit on that.

On the rocks it would rip an already stressed and compressed hull apart, and on mud it would act like a suction cup holding the boat on the bottom.

If they do, is it noisier because of contact with the sub and sea bottom?

even in Sub rescue exercises the boats don't tend to bottom, the only boat that im aware of that used to drive along the bottom would have been NR1, the Russians have probably got something similar.
If a boat was on the bottom you would likely hear the metallic scraping of metal against rock (worrying) or the disturbance caused by a large stationary object against a current flow.

However this is one for the sonar bouys here (see what i did there) they will be able to clear that up ET3R is a good chap and good source for that its his field.

If a sub is not resting on the sea floor how do they maintain a steady
position with currents and such? Do they use maneuvering thrusters of some type?

I believe the Ohio and Vanguard classes of SSBN are capable of hovering at depth in order to fire missiles, so id assume theres a system there.

Would maneuvering thrusters make a lot of noise detectable by the enemy?

low horse power low speed, but high speed screw smaller prop possibly but we know the project 705 has electric creep motors similar to thrusters on each end of the aft diving planes, at speed they were noisy but at slow speed not so.