quick answer:
Scopes are useful for two things 1. verifying that your device is working as expected, normally this also involves a signal generator and inputting a set of known waveforms and seeing what comes out and 2. troubleshooting. The issue with multimeters is they can tell you that Ac voltage but NOT what the waveform looks like, for example is it noisy, has the top and or bottom been squared off due to clipping distortion etc etc. with the scope you can work backwards from the output and work out where things are starting to go wrong in the circuit...
for example looking at a square wave output you can check for instability (ringing on the top and bottom edges of the square wave), bandwidth, (a rounding of the corners of the square wave) etc. using a triangle wave you can easily see clipping of the output (loss of the sharp top and bottom of the triangle wave) etc. In other words you need a signal generator to provide test signals for the scope to be able to see what is going on.... You can look at the outputs of DC power supplies and look for noise. ripple, start-up overshoot, undershoot etc. A scope allows you to dynamically see changes in a way a multimeter simply cant convey.
The probes decide the voltages you can measure (most scopes internally can only handle low voltages typically a few 10s of volts) . For example a 1x probe passes the voltage unmodified. A 10x probe divides by 10, a 100x probe divides by 100. probes themselves will have a voltage rating - which will depend on if they are 1x, 10x etc. Some probes are switchable between multiple divisors e.g. x1, 10x switchable.
The main issue with most scopes is that the ground lead is earth referenced (this is not the case for almost all multimeters) so if you clip the ground lead to something in your device which is not ground then you have just shorted that thing to ground and this can kill the device and the scope as well as being dangerous to the user. You can buy differential scope probes which don't suffer from this danger but these are far more expensive than normal probes and far more noisy. A battery power scope would normally not be earth referenced and so be fine without differential probes...
4 channel scope is a luxury, it enables you to look for example at both the + and - outputs of both channels of a stereo a differential pair, or the output of one channel and the input at the same time. A 2 channel scope is ok but to check output against input on both channels you will have to unclip the probes and move to the other channel.
I have the rigol ds1054z... its an oldish design but it quite good value for money for its features... especially if you can get a deal with the extra software options enabled for free e.g. more memory, serial bus decoding etc.
If you dont want to spend 1054z even an old second hand analog scope will be useful - just most dont make automatic measurements and require a bit more skill to use and read the screen to measure manually
do you have a signal generator? either software you can run on your computer and output from the sound card or an actual standalone signal gen? without one a scope is of much more limited use... Some scopes have the option of a built in signal gen but with some you pay more than a standalone gen, lose some versatility etc. So dont assume built in is cheaper.. check.
For audio you probably will not need a bandwidth higher than a few 100Khz both for signal gen and scope. Almost all scopes go into the 10s of mega hz or higher so that's not a problem. Cheap scopes will not be able to measure distortion or signal to noise ratio etc or give you a good FFT analysis of the harmonics in a signal but this is only necessary if you are benchmarking, developing, debugging or comparing etc.
If you are more serious about audio quality measurements there is a keithley 2015thd multimeter which has a built in signal generator and can do distortion, fft and noise measurements but its quite old, many of the advanced features require talking to it via a program and they are grossly overpriced on ebay. I managed to get one before they become crazily expensive and by modern standards they are behind the curve for example the signal gen is only 10hz to 20khz, the fft has a maximum of only 1000 bins and can't analyse bellow 20hz or above 50khz. it cant calculate distortion + noise for a fundamental frequency bellow 60.8hz etc etc. The user interface is not great for the advanced features and programming it is a pain (I have written some control software for it but its in beta and desperately requires more testing and refinement). I wish someone would make a modern version of the 2015 but i guess the potential market is just too small....
Don't connect the output of a stax amp to the input of a line level sound card , audio precision analyser etc.to try to do some measurements these devices are only designed to take a few volts input and a stax amp can fry them really easily.
If you need a good multimeter I highly recommend the brymen bm869s it can handle very high voltages in any mode. For example, I by accident had it in resistance mode and connected it across a 350V DC power supply. Multi meter survived no problem.. its very robust... and the dc power supply fortunately was robust and had current limiting and also survived. the brymen will handle multiple KV across any input: (Joe smith does some comprehensive multimeter testing)