Is there a general vertical speed that pilots shoot for, or is it entirely by feel/trial-and-error?
I would argue that the way to think of the perfect landing is not in terms of vertical speed but rather in terms of the G-forces involved.
Thinking in terms of G-force, the perfect force depends on what you're landing on and how long it is. The various conditions I can think of offhand are:
long dry runway, no wind or straight down the runway -- try for a greaser if you want, although arguably a greaser is not the safest landing.
cross wind -- the higher the xwind the more you want a firm touchdown.
short runway -- firm so you will have effective braking.
wet runway -- firm, especially if the runway has standing water. The two times that I blew tires on landing was when there was standing water and I didn't plant it firmly enough. In the first case the tires blew before I used the brakes. It was an F27 and one of our pilots was in back at a window seat. He said the two right wheels never rotated at all. I had touched down with a greaser on a puddle.
carrier -- I once checked out a Navy carrier pilot in a 182. It was really hard for him to flare for a landing, since they don't do that. As I remember, he said that they wanted a 4G landing impact.
Are you talking the approach-to-land, or actual landing? Keep in mind, when you flare, you are technically arresting a descent, so the VSI should not read much more/less than zero. For a stable approach, 500-700 FPM on a 3 degree glide slope usually works out well for ILS approaches. Keep in mind, all instrument approaches terminate visually, unless you are in a CAT III capable aircraft, which I do not fly.
In my experience it's feel/trial-and-error-and-error-and-error.
5 feet off the runway is not a good time to be looking at your VSI.
Since you're looking at a record of your vertical speed after the fact:
Lower numbers are better of course. The goal is to get the rate low enough that the landing gear can absorb the impact without passing on any of the energy. 60 FPM would be 1 foot/second. I'm sure most landing gear would soak that up and earn you praises from your passengers.
I've had exactly one "squeeker" - I didn't know I'd touched down until the wheels started moving. Naturally, I was in a single seat airplane with no witnesses :-(