how many cases are there actually of Eevee evolutions being depressed from regretting their chosen evolution? It's a fairly common thing in fiction, but how often does it actually occur?
Though fiction does portray it as something that most often affects eevee, post-evolution depression can occur in any pokémon. However, it’s actually a pretty uncommon occurrence. Sometimes it may be brought on due to changes in a pokémon’s hormones, or it may be due to difficulty in adjusting to new abilities; pokémon that change or gain a type often struggle for a short time after evolution, which, depending on the pokémon, can cause depression-like symptoms. Other times, yes, a pokémon may simply regret evolving, or (in eevee and other pokémon with multiple options) wish it had chosen something different.
Either way, the best thing a trainer can do for a pokémon that seems to be struggling with PED is be there for it. Reassure it that it is loved regardless of its form. If it is willing, start training it; not only is exercise important for a pokémon’s mental and physical health, training will help it adjust to its new body and abilities, which will then help it feel more comfortable as it is. If it doesn’t seem to be improving after enough time, or if it refuses to eat or train, consider taking it to a pokémon centre, as most offer psychological help as well as medical.