The iOS version of the Instagram app uses Freight Sans for large type (usually centered headlines) and the system’s Neue Helvetica for everything else. On Android, that system’s Roboto is used along with Freight.
The Instagram website uses Proxima Nova for all text with Neue Helvetica as a fallback.
The current logo, which replaced a sloppy Billabong-based mark in May 2013, is custom lettering by Mackey Saturday. The icon sports a little “Insta” badge in Chunk.
In these samples of the old Instagram app, that info uses the OS fonts from Android or iOS (depending on the device). So in 2012 that would be Roboto or Neue Helvetica (replaced by San Francisco in today’s iOS).