Flying dreams are common. In a survey of 1,181 Canadian university students, 48.3 % said they had ever dreamed of flying or soaring through the air, men (58.1 %) more often than women (44.4 %). Along with being chased and falling, it is one of the dreams people most often name as their earliest.
In the Typical Dreams Questionnaire study (Nielsen et al., 2003), "flying or soaring through the air" ranked ninth of 55 themes at 48.3 %. Men reported it significantly more often than women (58.1 % vs 44.4 %, p = .00002).
The figure counts people who have ever had the dream, not how often it happens.
Does flying make you realise you're dreaming?
It's often said that flying triggers lucid dreaming. One small, older study (Barrett, 1987: 56 students, 11 flying dreams) found the reverse order: "the lucid state preceded the flight". It is too small to settle the question.
Questions
Is it normal to dream about flying?
Yes. Nearly half of students in a large survey said they had ever flown in a dream.
Are flying dreams lucid dreams?
Not necessarily. In one small study, people became lucid first and then flew, not the other way round. It was too small to be conclusive.
Sources
Nielsen, T. A., Zadra, A. L., Simard, V., Saucier, S., Stenstrom, P., Smith, C., & Kuiken, D. (2003). The typical dreams of Canadian university students. Dreaming, 13(4), 211–235. https://doi.org/10.1023/B:DREM.0000003144.40929.0b
Barrett, D. (1987). Flying dreams and lucidity: An empirical study of their relationship. Lucidity Letter, 6(1).
73.8 %of students said they'd dreamed of falling
81.5 %said they'd been chased: the top dream of 55
18.8 %said they'd dreamed of losing their teeth