In 5e, the rules don't give a straight answer to your question, but Aarakocra most likely uses the Dwarvish script.
Aarakocra are described as originating from the Elemental Plane of Air:
They hail from a world beyond—from the boundless vistas of the Elemental Plane of Air.
Elemental Evil Player's Companion, p. 3
Furthermore, they "speak, read, and write Common, Aarakocra, and Auran".
Auran is described as a dialect of Primordial:
Some of these languages are actually families of languages with many dialects. For example, the Primordial language includes the Auran, Aquan, Ignan, and Terran dialects, one for each of the four elemental planes.
PHB, p. 123; emphasis mine
On the same page as the above quote, we can consult the "Exotic Languages" table to discover that the Primordial language is spoken by Elementals and uses the Dwarvish script. Auran is the primordial dialect for the elemental plane of air, as can be gathered by the fact that air-based elementals (e.g. MM, p. 124) speak Auran.
Now, while "Aarakocra" is not "Auran", Aarakocra speak both languages. I believe that in the absence of a RAW answer, it's safe to assume that their language most likely (!) uses the same script as the Auran Primordial dialect.
Furthermore, Aarakocra does not seem to have been planned as its own language at the start of 5e - the DMG (which came out before Aarakocra were released as a player race) has guidance for creating Aarakocra NPCs, and it only gives them Auran as a default language.
Therefore, Aarakocra most likely uses the Dwarvish script.
Assuming that all scripts that exist (excluding irrelevant ones invented by individuals) are listed in the PHB on page 123, we have Common, Dwarvish, Elvish, Infernal, Celestial and Draconic.
Out of those, besides Dwarvish, Common is the only other option that makes any sense. However, considering that Aarakocra in the MM (p. 12) don't speak Common (unlike player character Aarakocras), I'd still go with the Dwarvish script.