While Instagram's parent company Facebook has been quick to adopt support for 360 videos and photos, Instagram has thus far not supported 360 photos or videos yet. However, a recent interview with Bloomberg suggests that Instagram is working on adding support for spherical 360 videos and photos (sometimes called "VR").
Instagram CEO and co-founder Kevin Systrom spoke to Bloomberg's Emily Chang. During the interview, they discussed among other things VR's impact on Instagram. Here's what Systrom had to say:
"Well if [Instagram's] vision is to make you feel like you can travel anywhere in the world in a fraction of a second and experience whatever's happening in the world, imagine the day when you can put on a headset and be at a Coldplay concert seeing something happen like a big protest or a big riot anywhere in the world or something as simple as a friend's wedding. That's the kind of experience we'd love to create and I think that virtual reality in the coming years will play a critical role in seeing that vision come true." (emphasis added).
From Systrom's statement, it appears that Instagram is indeed working on supporting spherical 360 photos and videos -- the kind viewed on a VR headset such as Samsung Gear VR, Oculus Rift, HTC Vive, or even the Playstation VR.
Here is a link to the interview (around 11:20).