Nothing in the universe stands still.
This is not quite correct.
Every object is still (i.e. at rest) in it's own reference frame.
A black hole follows a trajectory through the universe, moving from one locus in spacetime to another locus.
Except that the black hole is at rest in it's own frame of reference.
Deep inside each black hole is a point that we've never been able to characterize.
We do characterize it as a singularity.
My question is, does the singularity, and possibly the powerful gravitational fields around the singularity have any measurable effect on the area of spacetime it passed through ?
The singularity is a property of the space-time distortion inside a black hole, i.e. a result of the gravitational field of the black hole.
The gravitational field has effects on space-time both inside and outside the event horizon. The event horizon is what defines the black hole's extent.
One imagines an undeformed area of spacetime. After some time, a black hole passes through this spacetime, then moves on. Does the spacetime it passed through simply go back to being everyday, garden variety undeformed spacetime?
It becomes more and more distorted and then less and less distorted. As the range of the gravitational field is infinite, it never actually is undistorted.
There is no permanent effect on space-time (e.g. nothing like a track "goughed" through space-time).
However, as the space-time is distorted there is a kind of permanent effect in the sense that e.g. an observer could see an object passing through such a gravitational field as experiencing a time dilation relative to the (distant) observer. That would be permanent in the sense that the difference in the time observer and object calculate is not going to be undone. The rate at which time changes will return to normal, but not the accumulated time they each record, which will remain different (in the absence of the observer passing through a similar distortion in space-time).