Test schedule is dependent on available gear. I generally have a nasty little analogue multimeter in my shoulder bag at all times, a miniature digital multimeter whenever the battery's not flat (it enjoys turning itself on without me asking it) a nasty little transistor amp thing with about a 11/2" speaker, and an XLR to minijack with balancing transformer in it, and a Swiss army penknife. 'Course, if I know I'm going out to mend things I've got a signal generator, soldering iron, a scope, a film can of often required components, spare fuses, an audio test CD… but it's surprising how much you can do with the basics.
Plug a powered speaker into a working output (say aux 3). Plug a signal source (preferably hands free, an mp3 player or CD; a microphone confuses matters. However, it does work.) into an input of the mixer, then turn up relevant volumes until you have a comfortable, not too loud, signal out of the speaker. Take a rubber hammer - not got a rubber hammer? - take an ordinary hammer and a piece of scrap wood, rest the wood against the side cheek and give it a firm tap with the hammer. This is not a racial slur, we're looking for bad contacts, dry joints, even cracked PC boards (although if we find those the time has possibly come to put the machine in the hands of an experienced service technician, even if it costs some.), the sort of mechanical damage touring consoles often develop. If there are no clicks, clunks or pops at this point we repeat the operation on one of the defective outputs. This will quite likely give a reaction, but unfortunately this is true whether it's dry joints on the output XLRs, badly seated connectors or just about any cracks or internal shorts. In fact, seeing that the thing has once worked, we'd be more surprised if this didn't give some reaction.
Looking at a picture I can't see how to open the thing; it seems to be a continuous face plate. Which is bad, because the next step is to open it up and give it a meticulous visual inspection. It's unlikely from the symptoms that it's a power supply problem (but that doesn't mean impossible) look for discoloured components, bulgy capacitors, badly seated connectors, solder joints with a mat finish rather than shiny (oh, and the 'blanket solder' comment was heat up all solder joints to connectors, even if they look OK; XLRs have a higher thermal capacity than resistors or ICs, and are more likely to make a poor joint with flow soldering). At least at a first stage do this with the power connector out; later you'll probably find me propping it open and going round the interior with probes, wiggling things, but I've been doing this for ages. Close it, plug it back in, and see if the problem's changed (either way).
Cyclecamper :- If the manual I've downloaded is right, no inserts on the auxes, only on the subgroups.
JM Fahey:- I do indeed distrust Soundcraft ribbon connectors; in older consoles (when they were a bit more modular than this one ) I've gone in and found bus connectors completely dropped out of sockets. I suspect employees of not having inserted them any too well originally, as they seemed pretty firm when well pressed home. However, I've never had pops; just silence when there was supposed to be signal. However, I will certify that one end can drop ou, making it not immediately obvious, and losing some functions but not others.