Like Uniballer said, you really need a powerful scope to work on "slow" digital stuff. Digital is square waves, which are many, many overtones, octaves higher, than the analog equivalent sine wave.
A logic analyser is a whole different animal, and yes, you could use one of those more than either a logic probe or oscilloscope. I've never owned one, and I'm not sure if I ever used one. I do mostly analog work. But, when I work on computers, I've got this cheap, no-brand-name logic probe that I bought in 1994, which always gets the job done.
I've used other logic probes that don't work well for pulsed circuits. In fact, everyone I've used other than my own, I've found useless. I don't know where to get one like mine, I've tried to find another one. But, every time I get one that looks like it, it doesn't work as well.
I've never had trouble picking out pulsed signals at 20MHz. I often have to take multiple readings, and compare the audio tone to a logic high and logic low, but I always get an idea of what is happening on a circuit. It has a Pulse mode where it only chirps once when the logic state changes. So as the state goes from low to high it chirps, and when it goes from high to low it chirps again. And, it lights green and red LEDs to reflect the state. Based on the distinct modulated sound, and the apparent colour of the two LEDs alternating at high frequency, it's usually pretty easy to tell if a circuit is doing what it's supposed to.
Obviously, if you need to compare timing between two circuits, you need at the minimum an oscilloscope, but, more useful would be a logic analyser. I haven't looked at those lately, but I would assume the way things are today that they are affordable. If you can afford one, go that route instead of a scope, if you don't plan on doing a lot of analog work.
I've never found myself in a position where I absolutely had to compare two circuits at high speed. I'm not saying I never will, but I haven't so far.
Just recently, I went through a lot of trouble to set up my scope, to work on two Amiga 2000s. But, my very high quality 100MHz scope is pretty useless with a 7MHz clock. I ended up abandoning it and using my probe, as always. I've got a 200MHz scope that I probably would have had the same issue with.
