I own a 78 Fender Musicmaster. Mine is all original, in pristine shape when I bought it, and it still had the original chipboard case that was disintegrating all around it. I really like the bass and I have used it for gigs without problems.
Things I would note include:
My MM is pig heavy! The damn thing weighs more than my 96 Am. Std. P Bass, but it balances perfectly.
Watch for a cracked or damaged pickguard around the jack. This is a weak spot when pulling out your cable, so you will need to be careful, otherwise it is easy to do damage.
With the original saddle, perfect intonation up and down the fretboard is not possible. This is not a big deal, of course, but it is something to consider.
The fretboard radius is not as flat as more modern basses (or at least it doesn't feel as flat). It is probably a 7.25" radius, but I could be wrong.
A lot of bassists complain that the MM has low output from the pickups. I find that I can work with my amp to get the sound I need without problems. Honestly, mine doesn't sound that "thin" at all. It has more heft than one Jazz bass I own.
Fender's Mustang Bass case will fit it perfectly, if you choose to use a case. This was a good investment for me because I like to have cases for all my instruments when going from gig to home to gig.
All in all, I really like it. It is my only short scale and it sounds really good played with a pick!