Several of the later AF and early AF-D families of Nikkors released in the 1990s were good lenses. I still use these two regularly:

• AF Zoom Nikkor 35-80mm f/4.0-5.6D . If you can find enough light to shoot with this lens, it turns in sharp, contrasty images. I pull this one out when I'm going to be shooting outdoors and don't want to juggle primes or the bulk of my larger glass. You want the Mark I version of this lens, which has a distance scale marked on the focusing ring, preferably one that was made in Japan. The Mark II and III versions are a completely different design that's nowhere near as good optically or build-wise. I have one each of the Japanese- and Thai-made Mark Is, and my (subjective) opinion is that the Japanese copy turns in better work, but that could just be the luck of the draw with the two I got. These were kit lenses for a number of N- and F-series film bodies and can be had for under $100, sometimes with the body still attached.

• AF Zoom Nikkor 28-105mm f/3.5-4.5D. This one has a permanent slot in my bag because it's reasonably sharp, provides good contrast, doesn't add a lot of distortion and works well as a macro lens. These run around $200 used. EDIT: I tried the 24-120 before buying this lens but didn't care for its flavor of distortion and found the colors a little washed-out. YMMV.

I've used both lenses on full-frame film bodies and don't remember either having vignetting problems.

If you want something a little faster, you might see if you can find a clean 35-70 f/2.8D. I've never owned one, but they always received high marks from those that did.