One of my players is a scourge aasimar. I see absolutely no indication anywhere that he can fly but he created the character thinking he could, indeed, fly. At this point, he's level one and hasn't played the character yet. Can scourge aasimar fly?
Only the Protector Aasimar have flight
The Radiant Soul feature of the Protector Aasimar offers 1 minute of flight at a flying speed of 30 per long rest. (Volo's Guide to Monsters, p. 105).
The Scourge Aasimar has a light / radiant-damage-causing racial effect (Radiant Consumption), and the Fallen Aasmiar have a fear effect (Necrotic Shroud) - their wings are described as "flightless" in VGtM.
Neither Scourge nor Fallen aasimar has flight as a racial feature.
(FWIW, all of this is on VGtM, p. 105)
If you are playing Adventurer's League, then yes, during the current season (9) every Aasimar and Tiefling can trade in specific abilities for the flight ability. The Aasimar loses the Light cantrip and the level 3 ability, the Tiefling loses the Infernal Legacy (or replacement) trait, but they get permanent wings in return, adding an innate flying speed of 30ft as long as they don't wear Heavy Armor.
To expand on KorvinStarmast's answer, the Scourge Aasimar (VGtM, p. 105) has no explicit racial features that grant flight. However, if your player wants to keep his character's subrace at level 1, desperately wants wings, and is patient enough to let the story unfold, you could turn a problem into an opportunity for interesting story and character development.
Aasimar "subraces" are flexible in the same way that Paladins often change their Oath subclass based on their character choices. Consider how the description for Scourge Aasimar includes the following:
[Scourge Aasimar have] a powerful desire to destroy evil--a desire that is, as its best, unflinching and, at its worst, all-consuming.
Scourge Aasimar, while not truly fitting the "fallen angels" trope of the Fallen Aasimar, do have some undertones of "angelic imperfection" that separate them from the Protector Aasimar.
In order to allow your player to change their Aasimar subrace, you could introduce a quest wherein the Scourge Aasimar either seeks redemption or to better fit the ideal of their Angelic Guide, thereby becoming a Protector Aasimar and gaining the attribute of flight. This would be very similar to the way that an Oathbreaker Paladin is redeemed, becoming a true Paladin.
For lower level campaigns, this could be beneficial as well, as flight is often cited as a source of early-game imbalance.