sometimes mispelled "analyzer"
This must be one of those "british" things; what is interesting is, that if put into Google, it will ask if you wanted "analyzer" if you put in "analyser"; but the first hit for "analyser" is a wiki article, with a sub-hit wiki article on "spectrum analyzer"...
bah - spell it how you want; maybe look both up!
As far as your question is concerned, tkbyd, as others have noted, many people have done this. I was just looking this up a few days ago, and found this interesting version:
http://www.arduino.cc/playground/Main/LogicAnalyzer
Which seems to do direct-port reads using inline assembler in what is known as an "unrolled loop" fashion for maximum speed (only 1024 samples, though - but of 8 pins per sample, it looks like?).
Many other examples can be found - also google "arduino logic analyzer" for several hits.
Something to realize is that a logic analyzer is not necessarily the same as an oscilloscope; they both measure waveforms, but the logic analyzer is more concerned with a number of waveforms (digital square waves) and their timing in relation to each other, while an oscilloscope measures characteristics about the wave and its form itself over time (perhaps in relationship to other input waves as well, though once you get past 2 channels, scopes get really, really expensive - I've seen some four-channel scopes that would have you mortgaging your house; most oscilloscopes you see though are 2 channel max, with maybe a third "Z" channel on occasion - generally for triggering).
As you can tell, you can build such a scope (logic analyzer or oscope) with an Arduino, and there is enough speed available to make it useful for sub-MHz measurements; you might even be able to approach the MHz barrier if you set certain limits and write very tight assembler code (I would look at Udo's example above as a model), but ultimately the ATMega8/168/328 is limited for the task.
Things get much more interesting though, as you move up the ladder of Atmel's offerings. For instance, the JYE Tech oscilloscope uses such a better processor (can't recall which):
http://www.jyetech.com/Products/LcdScope/eDSO062.php
It has (supposedly) 1 MHz of bandwidth for $49.00, which is a steal for a single channel scope. For most hobby uses, this is enough. Plenty for checking servo timings and the like.
If you want to go beyond this, though, you have to spend some money, unfortunately (though there are plenty of options available under $500.00 - both standalone scopes and USB scopes - in the 10-40 MHz range).
or even a full blown digital