If you complete the quest's desert leg first and speak to a Goron gentleman not once, but twice, the game locks out completion of the other two portions, halting all progress in the game. That said, it's pretty easy to avoid: just don't talk to the Goron more than once. Done! (Alternatively, do not complete the desert part first.)
Nintendo gets more specific in describing the glitch in its letter, which we've dropped below the break. We've reached out to Nintendo to see if a patch will be issued for the game, but the company's official release unfortunately didn't indicate as much.
The situation is caused by performing the following sequence of events during the Song of the Hero quest:
1. At the beginning of the quest, go to Lanayru Desert to retrieve the song of the Thunder Dragon.
2. In the Lanayru Mine, speak with Golo the Goron.
3. Complete the Thunder Dragon's event, and receive his song.
4. Before heading to the forest or volcano regions, speak with Golo in the mine again.
5. At this point, the forest and volcano events will no longer occur, making it impossible to continue.
• This issue WILL NOT OCCUR if the Fire and Water Dragon songs have already been collected before collecting the Thunder Dragon's song. Talking to Golo in the Lanayru Caves will also NOT trigger this issue.
• If the game has been saved after completing the sequence of events, it will no longer be possible to proceed through the game on that save file. The only options at that point are to either start the game over or to use an earlier save file to continue.
I wish I had a better answer, but I'm afraid there currently is no other fix available. I can assure, however, that we are documenting all feedback we receive on this issue. Thank you for your patience and support.