While “[n]o spell yet devised enables wizards to fly unaided in human form” (QA1), various spells create flying effects.
References from the canon
- The Levitation Charm ('Wingardium Leviosa') levitates objects (PS10).
- The house-elf Dobby used a Hover Charm to float a pudding above Aunt Petunia's kitchen (CS2).
- Spells cast on broomsticks and carpets allow them to fly.
- Lord Voldemort learned to fly without a broom, much to the surprise of the Order of the Phoenix, who discovered this ability during the Battle of the Seven Potters (DH4, DH5).
- Another person known to fly without a broom is Severus Snape, who learned the magic from Voldemort (DH30).
- By studying her father's skills, his daughter Delphi taught herself to fly without a broom (CC3.20, CC4.11).
The films show Quirrell (PS/f) and Death Eaters (OP/f, HBP/f, etc.) flying through the air. This effect, while effective visually, is definitely not canon.
Inconstistency: The book Quidditch Through the Ages was written long before Snape and Voldemort learned to fly without brooms, which is why the inconsistency about there being "no spell known" to let wizards fly without broomsticks.