Does an Oscilloscope's bandwidth need to match or exceed the microcontroller's/processor's clock speed?
The bandwidth specifications on an oscilloscope are for sine waves only. For example, if your oscilloscope has 100MHz bandwidth that means that it won't have any problem with 100MHz sine waves. 100MHz square waves are a different matter - generally to get a good look at a square waveform you'll need 10x the bandwidth. So, for a 100MHz square wave to show up nicely on your oscilloscope you'll need 1GHz bandwidth.