For a transport aeroplane there's a document called the MEL - Minimum Equipment List, which defined the stuff that has to be working.
For e.g. training aeroplanes, the document trail is a little vaguer, but ultimately a decision is made by the Captain, or if a student is flying it, the club, as to what is considered essential.
There are however numerous cases where it's perfectly reasonable and safe to fly an aeroplane, even in the training environment, with unserviceable instruments. You don't for VFR flying need many of the IFR instruments (AI, T&S...), for much flying the compass will serve without the DI being serviceable, very often the second altimeter is only a "nice to have".
Engine instruments are again subject to opinion to some extent. EGT isn't strictly required on most engines, whilst similarly unless you have a variable pitch prop, an MAP gauge is no more than a nice to have. On the other hand, I'd not fly many - if any - 4-stroke engines without a working oil pressure gauge, for example.
On the other hand, if instruments are fitted, I'd always much rather have them working than not! In a well run club, insisting on a high degree of serviceability is thoroughly sensible.
G