I've discovered that it is showing messages that contain shared music, and not messages with just text. Is this normal? And if so, why can you not have conversations over the mobile app like you can on the computer app
From what you're saying, it sounds like there is an issue with your particular situation, I recommend reaching out directly to Spotify Support. They can be reached through their Twitter Page @SpotifyCares or through the online contact form.
Make sure to check your inbox and junk folder for an email from Spotify.Once you get it, make sure to reply back right away, even if it says it's from a "no-reply" email. This will assign your case to a Spotify rep and you should hear back within 24 hours.
Thanks again, I will do this as soon as their online request forms are up, it says to wait 24 hours. I went on my ipod and it did the same. Are you sure I can receive all messages on the mobile app, or just ones that are linked with sent music? Because both are doing the same thing, despite being on different devices.
The inbox on the mobile applications is still the old inbox system, not the new messages system like on the desktop. That means that unfortunately this isn't a bug as the inbox screen was only designed to display tracks that have been sent to you, not the conversation view which is shown on the desktop.