The content of ephemeral social media like Snapchat may stick around for longer than you'd guess. The basic concept behind Snapchat is that messages will be automatically deleted after they have been viewed. Once all recipients of a Snapchat message have viewed a message it is deleted from the Snapchat servers. As the policy posted here makes clear, messages which are not opened will be deleted 30 days after they are sent.
When two users communicate in a one on one chat. their messages will be deleted when the chat ends, but the settings can be changed so these messages are retained for 24 hours. Snapchats can be saved by simply pressing on a chat and holding on it until a Snapchat's background turns gray.
Snappables are inactive games multiple users can play in Snapchat. These can saved by a user in the memories archive indefinitely. The online policy makes clear that Snapchat backs up snaps saved to 'Memories'.
Snapchat also hosts online 'Stories' which are public content. This content may remain after a user has deleted their post.