Q: Would I be able to write a musical titled, “Danny Phantom the Musical” and use the likeliness of the characters?

A: The biggest thing that you have to worry about is probably copyright because you would be making a derivative work by making a musical. Viacom has the registration for the show, and when you adapt and derive content from one medium into another, you need a license from the respective owner(s). Otherwise, it's a copyright infringement, most likely.

For copyright, it also depends on what all you're doing/how you're using the characters. There are possible fair use considerations, but as a reminder, fair use is a legal defense and not a legal right in America. The judge/court will be the only party able to confirm something is a fair use, so also keep that in mind.

As for trademarks, it will come down to whether or not you either (1) use the characters in commerce and as a designation of origin or (2) whether your potential use is considered dilution or tarnishment of the existing trademark. Trademarks are source identifiers, and if you're not using the trademark as a brand, then you may not be infringing any trademark rights. If you're making money from it and they find out, then I'd say expect a cease and desist letter at the least.

If Viacom believes that your musical with the Danny Phantom characters diminishes their trademark rights, then they may try to sue for dilution/tarnishment, especially if they try to suggest your musical is low quality.

These are all some basic possibilities to keep in mind. Generally, if you're going to use a referenced work, then asking and receiving written permission is the best way to go. If you start work on the musical and hear from Viacom, then I absolutely recommend contacting an attorney ASAP.