I would like be able to identify problems in a mix by looking at the spectrum analyser, now I realize that mixing is done with the ears, not the eyes but I mix in a room with bad acoustics so I cannot rely only on what I hear. How do problems in a mix look in a spectrum analyser? (by problems I mean things like resonances, mud or anything that can make a song sound bad)
We ALL mix in a room with bad acoustics (and using imperfect monitors)! That's why we get nice and close to our monitor speakers - more direct sound, less room sound - and why we take our mixes to different playback systems - car stereo, home hi-fi, phone and earbuds, boom-box - to check what a range of listners will hear. With experience, we get to know how a mix needs to sound on our studio monitors.
There's one thing a spectrum analyser CAN be invaluable for. It's easy to over-do the low bass. YOU may not have a playback system that goes (audibly) down to 20Hz... (The flip-side of this is that sometimes 'compatibility' isn't the way to go. If you're making EDM aimed at club systems, do a 'Club mix' with room-shaking bass, a 'Radio mix' without.)
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Long story short: resonances are frequencies with especially high amplitude (so look at especially bright stripes on spectrum analyser), mud is too much lower frequencies (and that is not so obvious to tell by only looking at the spectrum).
But in my opinion it is impossible to make a good mix just by looking, you REALLY need to HEAR it. If the acoustics of the room is a problem no worries - just equip yourself with proper headphones. Better mix on the headphones than on monitors in room with bad acoustics. I recommend for example Sennheiser HD600, they have lot of detail and do not enhance the sound (the characteristic is flat as possible). Of course it's not the only model, you should browse more audio engineering-related forums to find other recommended models.
Also, please use search function, there was quite similar question recently - How to identify mud and resonance in a mix
I use a spectrum analyzer to equalize the sound in the room, the monitors and the room will have their own resonances which can be identified with the analyzer. When the peaks are attenuated and the weak frequencies boosted to achieve flat response in the room then the listener/engineer has the most ideal possibility of adjusting the mix to accurately represent the original sound as it was recorded. That said, spectrum analyzers and equalizers have their limitations, well designed rooms and flat responding monitors are also very important.