For Norse mythology, there isn't a focus on winged animals so much as there are just versions of ground animals who can fly. Flying horses are popular, with Sleipnir and Hofvarpnir being two examples. Thor also has a chariot drawn by goats which seem to be implied to be capable of flight. Valkyries also fly around, but whether they are capable of flight on their own or ride flying horses depends on the source.
There's also some instances of magic items allowing people (well, gods, for the most part) to fly. Freyja has a cloak made of falcon feathers that she uses to fly, which is occasionally loaned out to other gods in need of flying abilities.
If you want to keep with a Norse theme, these animals probably aren't a part of everyday life. Flying animals and magic tend to be tied to the gods and the giants, so it's likely that if a mortal hero gains a flying mount, it's a one-off gift from one of these sources, rather than something produced by mortal hands. Dwarves also produce magical artifacts, so something like a flying chariot or ring might come from them. In any case, these are usually rewards for heroic undertakings.