All of the Spielberg-produced cartoons from the 90s had boppin' themes. Animaniacs, Freakazoid, Pinky & the Brain... not sure those theme songs have ever left my head.
A lot of my favorites are from shows long past. The original Hawaii Five-O theme is absolutely prefect for the show. Mannix, Barney Miller, and Mission Impossible all had music that was great in its own right.
Somewhat more recently, the West Wing and Firefly themes were outstanding.
My favorites from the last few years are Game of Thrones and Daredevil. One of my favorite commentary tracks ever features the child actors who played Sansa, Arya, and Bran on Game of Thrones singing along with the introduction.
Song of the Bandits The song on the YT video isn't great here imo, but they actually mix it up in the show proper and use different songs. And some really fit the theme.
If we're talking about music in general through the show rather than just the intro - The Korean High Fantasy series Arthdal Chronicles. Some stunning, music, throughout, the show. It was also kinda like watching a JRPG (even if Korean).