There have been quite a few good suggestions, but there's a little I can
add.

I've successfully found shorts between inner layers of multiple-layer PC
boards due to misaligned layers. (When layers are misaligned,
plated-through holes -- vias -- can contact layers they're supposed to
miss, shorting them.) The method I used was to connect a high-amplitude
signal generator (one that puts out a few volts into 50 ohms) of about
50 kHz between the shorted layers. Then, I used a small coil as a
detector, connected to a scope. The coil was a few hundred turns on a
ferrite bobbin (essentially a ferrite rod), about 1/4" diameter and 1"
long. With it, I was able to trace the current from the "hot" signal
generator terminal to the shorted via -- the signal drops off pretty
quickly beyond the short. The via was drilled out to clear the short,
and connections to the intended layers made manually.

If the frequency is too low, the AC current spreads too much on a
plane-type layer. If it's too high, it won't penetrate through layers
which are over the conducting one. If you're not dealing with multiple,
plane-type layers, you could use a higher frequency and it should be
pretty simple.

A toroid, as someone mentioned earlier, is a poor choice for a detector,
unless you grind a slot (gap) in it -- half of a split toroid would work
fine. I tried a floppy drive head, too, but found that it had very poor
sensitivity. They're apparently intended to be driven by high current,
and at quite a bit higher frequency. The resolution would be good,
though, if you put an amplifier between it and the scope. I found the
bobbin coil to be adequate.

I used this method quite a number of times. Knowing how to find
inner-layer shorts does have disadvantages, though. The very first
prototype PC boards for the Tek 11400 series 'scope mainframes arrived
at about 2:00 a.m. on the day they absolutely had to be built, and all
had inner layer shorts. Guess who got called in to fix them. As I
recall, I managed to fix something like two out of the four or five --
just enough for that day's build. The rest had too many shorts to salvage.

The same technique can be used to find shorted components, although some
of the other methods suggested might be easier depending on the
circumstances and type of boards.

Roy Lewallen, W7EL