The Stable Trotters are an odd bunch of The Legend of Zelda: Tears of the Kingdom. This batch of musicians all have very... distinct personalities and also view themselves as a little more important than they probably should.
But that doesn't mean they don't have their uses, as they help you bring the Great Fairies out of hiding with their playing. There are a handful of quests that will have you help them to do this, such as Serenade to Mija.
Updated by Shane Black on June 12th, 2023: Added some more in-text links to relevant guides.
Start Serenade To Mija
You can start this quest by going to the Snowfield Stable, up in the northern part of the map. The Stable is just southeast of the Hebra Mountains, right next to the Orochium Shrine.
There, you will find the Stable Trotters playing their music right outside, near where you will find Penn if you are doing the Potential Princess Sightings! quest. Speak with the conductor to get the quest going.
Complete Serenade to Mija
It wouldn't be a quest with the Stable Trotters if you didn't have to help them with some wagon trouble, which is exactly what you will have to deal with first. They are in need of a roof on their wagon to help them travel in the snowy weather. Luckily, there's a construction site right by the wagon where you can use a piece of wood from.
Use Ultrahand to pick up a piece of wood and attach it to the two poles on the wagon so that it covers the entire seating area of the wagon.
Once you have done that, you can head back to the troupe and speak to the conductor to get the whole band in the wagon. And, of course, they need you to drive them over with your horse. You will have to use a harness, which you can earn by earning Pony Points through the Stables. When you have the harness on your horse, bring it over to the wagon, then attach it with Ultrahand.
Now, all you need to do is steer them over to the Great Fairy's Fountain, where you will have a map marker to help lead you there. You can also make out the lighting from it. Just be careful of some enemies in the area, and be sure not to take too sharp of a turn and knock the wagon over.
Eventually, you will make it and trigger a cutscene that shows the band bringing out the Great Fairy with their music. After that, the quest is over. Don't forget to enhance any armor you have with the Great Fairy.