It’s an interesting question…
Im no EE, but feel quite safe around these kind of devices, esp. modern ones.
generally, Id say we are in no more danger than anyone else, given these days most households are filled to the brim with electrical equipment (tvs, washing machine, computers )
power adapter fails is it dangerous? … any electrical equipment could fail in a dangerous way, e.g. it could suddenly catch fire, and burn your house down.
if device has a external PSU, you can see what that outputs… id expect this to be pretty low for most synths/racks - so not real worry, a small shock , or burn perhaps.
note: some devices, you might get a small tingling of electricity from… I think its fairly common for some devices to ‘leak’ a small amount of current to ground. (i think ‘how much’ leak is allowed is subject to regulation?!)
generally, you’d hope most things would have some kind of fuse, so that if excessive current is drawn due to a fault - it’ll ‘blow’ … id also hope to sell stuff commercial it has to meet certain safety standards.
if you start opening things up (*) , then obviously remove all power.
( I personally, include adding/removing modules for my eurorack in this)
synths/racks with internally PSU, have a high AC voltage in them ,
but Id assume in eurorack which are meant to be ‘opened by end users’, all PSUs will be enclosed, and there will be a fuse.
for things which are not meant to be open (except for repair by qualified people) then you’d have to take more care. eg. some units might have large capacitors which hold charge even when turned off. particularly older equipment - I doubt I’d go poking in something with vacuum tubes !
… but I think thats getting more into DIY safety which is probably OT.
of course, things can fail… life has risks…
in terms of precautions…
I go for normal recommendations for your house - smoke detectors, fire extinguishers, and turn off things when they are unattended.
one thing we might be more likely to be guilty of is running a lot of devices off of multiple point extension leads - again most things take very little current, but it could be dangerous with lots of devices esp. if its an unfused extension lead.