The underlying point is that degradation is not one process, and which one dominates depends on the condition you are asking about. In solution at refrigerated temperature the rate-limiting pathway is usually deamidation and hydrolysis; at room temperature aggregation overtakes them; frozen, the damage happens during the transitions rather than during the hold.
Adsorption to the container is a real loss at low concentration. For a peptide at 0.1 mg/mL in an untreated glass vial, single-digit percentage losses to the wall are plausible; at 5 mg/mL it is negligible. This is one of several reasons not to reconstitute to a very dilute working solution and store it.
A domestic freezer holds roughly minus eighteen degrees and cycles by several degrees on its defrost schedule, which for a lyophilised solid is entirely adequate and for a frozen solution means repeated partial melting at the surface. If you are going to freeze a solution, an unopened chest freezer is materially better than the compartment in the top of a fridge.
Deamidation kinetics for asparagine in peptides are well characterised and strongly sequence-dependent: the residue following the asparagine dominates the rate, with glycine and serine at the n+1 position accelerating it by an order of magnitude relative to bulkier residues. That is why two peptides in the same buffer at the same temperature can have quite different shelf lives.
Store solid, store cold, store dry, and reconstitute what you will use rather than what fits in the vial.