Nothing is inherently immoral about singing and making music for a career, and nothing is inherently wrong, even, with being famous. However, much like being wealthy, I would caution against seeking fame for its own sake. You can have both wealth and fame as byproducts of your quest to do good work with the gifts God gave you, and I don’t think that’s wrong, but seeking to “become something big” for its own sake can lead down a dark path. If the “love of money is the root of many evils” I’d say the love of fame, at least in our culture, is very similar.
I say none of this in a preachy way: I love to sing and make music too, and often have a cyclic love/hate relationship with the idea of “hitting it big.” After all, we are artists, and the bigger your audience, the more people your art reaches, no? I have my own YouTube channel in the interests of that very thing. So it’s the strange tension for the Catholic artist, that we know that the love of fame for fame’s own sake is poison, but at the same time the desire to get our work out there to as many people as we can is simply a desire to live out our gifts in the most productive way possible.
So I’d say it’s mostly about your motive: Do you want to be famous mostly so that people will shower you with praise, or because being famous is just a “tool” to get your music heard, and thus reach the greatest number of people with the gifts God’s given you? The former is pride, but the latter is not immoral at all. It’s a hard line to walk though, and I’m sure I fail it ALL the time in my heart, although my strong love of privacy does ensure that at least I have days when I do legitimately see fame as scary, not desirable. Yet, since as I said, an artist by nature desires an audience, I find myself taking steps toward the very sorts of things that could in theory, in a one-in-a-million sort of way, lead to my becoming known on a wide scale.
As others have said here, it’s extremely unlikely to happen anyway, but if it does happen, try to keep your heart in the right place. Pray that I always do the same.