Answer first: dry, cold, dark and still, with the lot code where you can still read it in a year.
Reconstituted with a preservative-containing diluent: refrigerated, conventionally treated as usable for about 28 days from first entry. Without a preservative, single use and discard.
Keep vials in their original carton or in an opaque box. Light exposure matters for tryptophan-containing sequences and for anything with a chromophore, and a closed box is free.
Two-to-eight-degree refrigeration and minus-twenty freezing are the standard storage conditions for lyophilised peptides in supplier documentation across the market.
Appearance does not indicate stability; a normal-looking cake can be substantially degraded.
Label the vial at reconstitution with date, concentration and units per dose.