This weekend I did some spring cleaning in my garage, and discovered I had a rat or two living laaaarge in there this winter.
So I bought and set a trap and sure enough, next morning a dead rat. I also see another scurry out from under the bench, so I set the trap again that morning.
We come back in the afternoon and I find the trap has been triggered, but no rat. The trap is located in the same spot where it was set, and has a tiny sliver of blood on it. About a foot away, there is another drop of blood about the size of a quarter. Then about 8-ft away, a larger pool of dried blood, maybe 2-3 quarters size.
So here is the weird part. No bloody tracks or trail between pools of blood, and no body. The second pool of blood is located further into the garage, away from the door and trap, away from the escape route. I checked everywhere on the floor, nothing. Now I'm worried the little bastard might have climbed up somewhere to die. 
It sounds like you probably had the trap go off "early" and just grazed the rats nose. They'll bleed like a stuck pig from that, but may only leave a puddle where they stop to rub and lick the wound, rather than a blood trail.
The good news is it probably won't die and stink somewhere; when a trap deals the rat a deadly blow it's almost always going to be in the trap. If it's a glancing blow, it's probably not going to be fatal.
The bad news is that first, if you see a rat or two, you almost always have more. There's probably a colony nearby; maybe in your buildings, or maybe somewhere else within 50-100yards. Rats don't spend much time a as singles or doubles long-term. Next, they're SMART little critters, and communicate in spooky ways. If one sees another in a trap, it'll spread the word through the colony, and you won't catch another in that type of trap. Spooky, and hard to believe, but true.
If you poison them, it can cause real issues as well. They'll often take the bait to a secluded spot in a wall, etc, and with a new food will often sample just a TINY bit. If they get sick or die, you're done; none of the other rats will touch the bait. I had a colony on my property, and resorted to some poison blocks placed in areas that I knew my dogs couldn't get to. The rats found the unopened container/bucket in my garage, chewed into it and helped themselves to the entire batch. Where did they take it? I had no idea, but a week or so later my german shorthaired pointer began bringing my wife and I chunks of in her mouth. Not cool, but she never got sick... I eventually had to resort to non-poison methods and found stashes of uneaten blocks under my house foundation.
There are a few tricks to getting rid of them with traps. Place your traps carefully, to maximize the chance of a kill and minimize the chance of a glancing shot or misfire. Then, check and remove any dead rats ASAP, before any other rats witness the carnage. I'd even consider wearing rubber gloves and cleaning the traps between kills so there is no scent of "distress"; rat scent mixed with blood, urine, etc... Put out multiple traps and try to catch a bunch over the first 2-3 days, before the colony gets wise to the threat.
It's also extremely important to kill the dominant female. If you kill the males or younger females, more will keep coming. If you kill the dominant female though, the rest of the colony will often evacuate the area.
Good luck! Rats are no fun...
