Hi Michael, 

I agree with Touraj A Farzani regarding testing your restriction enzymes are actually working. I usually do single digestions on 1 µg of your vector with each enzyme separately - if you see a shift in the band size on an agarose gel compared to undigested vector, then you'll know they're working fine. Considering you get no colonies on your negative control plate and relatively few on your vector+insert plates, I'd say it's probably not an issue with the restriction enzymes (if it was, you'd get a lot more background on your -ve control plate).

As a general practice, I perform my ligations overnight at 4°C, rather than using a quick method, as I've found I get better ligation efficiencies that way (personally). One issue I have had before and has drove me mad at times, is dead ligase buffer/enzyme. You can test whether those are working by setting up several test ligations with different combinations of ligase buffer and enzyme (try some aliquots from another lab or colleague). Instead of using your vector in these test ligations, add a single µl of 1 Kb Plus DNA ladder (or whatever DNA ladder you use that contains fragments a few hundred bp in length). Simply incubate at the optimum ligase enzyme temperature for 15 minutes to an hour and then run on an agarose gel; you should see that the smaller DNA fragments all ligate together and you're just left with DNA bands that are several thousand bp in size. If so, your buffer and enzyme works, if the DNA ladder still looks normal then either of these reagents isn't working properly anymore and that's your problem. A way to prevent this from happening in the future is to aliquot your reagents into smaller volumes when you first open the tubes, that way excessive freeze-thaw cycles won't happen. 

Best of luck with your cloning!