I don't know (and partially don't really care) about the terms and conditions that Instagram, Twitter or Snapchat have about saving or downloading their content to my iPhone but InstaSave's recent update makes it absolutely simple to save all these things without a problem.
Many months earlier, during the time of iOS 6, we had a tweak of the same name helping us download Instagram photos. Things changed since then. Instagram got updated, Twitter released Vine, a short video share service, and then a lot of things happened with Snapchat too. And oh, not to forget, Instagram Videos happened as a competition to Vine.
Saving photos and videos from Instagram became almost impossible. Vine kind of remained out of bounds till VineDownloader showed up in Cydia. And Snapchat of course had those issues with captured screenshots sending alerts to the sender. All that changes with one recent update.
InstaSave, the iPhone app that had initially been designed to help users save Instagram photos, has now been updated to add Vine to its cap. You can now download Vine videos directly on your iPhone and save them to the camera roll (or other album).
InstaSave also has the capability to capture/download Instagram videos (something that I haven't noted so far) and save Snapchat photos too. While it notes that saved Snapchat photos are not marked (no alert is sent to the sender), we're not so sure how long this will remain so. (On a side note, Snapchat has been mired in security breaches and problems of its own).
Till a jailbreak comes up for these very functions (or till the developers of old tweaks update theirs to iOS 7), InstaSave is the only option left to save Instagram photos, videos and Vine videos.
We've looked at other solutions like IFTTT which is complicated for saving and you have to create multiple recipes. Many web-based services exist but these can be cranky when accessed from your iPhone. InstaSave remains the easiest, in my opinion.