What are the regulations regarding plane spotting in the US?
Generally absent a specific prohibition you can take a photo of anything you can see with your eyes from public property (like a sidewalk). That includes airports, airplanes, bridges, tunnels, police officers, kittens, and unicorns.
(For more in-depth information check out Andrew Kantor's excellent article and Bert Krages' excellent one-page flyer on photographers' rights, or check out http://photography.stackexchange.com - they've covered airport photography, though not in any real depth, and I'm sure they have some street photographers well versed in the legal aspects of taking pictures.)



So what constitutes a "specific prohibition" on photography? Generally a clearly visible sign that says PHOTOGRAPHY PROHIBITED (the old Grumman facility in Bethpage had such signs on all of the fences where there was a decent view of the runway).

Absent such a clear prohibition you're free to take all the photos you want as long as you're on public property while you're doing it (i.e. "not trespassing on airport property"), but be aware that some law enforcement officials may not see it that way, and you may have to have some uncomfortable interactions with them.
In my experience most police officers are well versed in the law, though occasionally you'll find one who isn't. You're more likely to have trouble with a "rent-a-cop" security guard though.



If you want really good photographs (and you're not photographing at a major airport like JFK or O'Hare) you might want to speak to the airport management. At smaller airports someone - either airport personnel or a local pilot - might be able and willing to escort you airside where you can get some really great shots.
It helps in this situation if you have a body of work you can show (either published photos or art prints). The caveat is that many airports will also ask that you carry some kind of insurance while you're photographing on their property (and at larger airline fields that requirement can be substantial).