Instagram is definitely a mobile-first experience, but that does not stop people from using the web version to see the bigger picture. The company has slightly tweaked the user interface and gave it a cleaner look, but functionally it has been a wreck; with the exception of liking and commenting, nothing can be done on the web.
Today, Instagram rolled out search functionality, so you can search for photos by hashtag, location, and also look up people by username. Finally!
Unfortunately you still can’t access your inbox nor can you tag someone in a comment, but some improvement is better than no improvement.

