Simple task Mr. Axeman. The first thing I would do is swap breakers with another of the same rating- it could be a bad breaker.
The majority of shorts occur in the outlet boxes. Begin by removing every outlet from the box and examine the wires carefully. Pay attention to where the clamp in the back of the box holds the wire. People like to tighten these down so the wall would pull apart faster than the wire. After everyting is out and no wires are touching, flip the breaker back on and see what happens. If you are using an ohmeter you can isolate the short on the ohms scale by touching each black wire to the box (power off of course). If the breaker stays on then find the one hot wire. Connect it back to the outlet along with the other black & white wire that sends it to the next outlet. Continue this until the breaker pops and you have found the short.
If the breaker popped when you first reinstalled the power, then the short is on the first leg of the run. Pull the black wire off the breaker and check that side of the wiring also. Do this and let us know if you still have problems. I am going to ass u me the meter you used was an ohmeter, even though I shouldn't.