Good question. I want to make several points...
1. Since there isn't actually such a thing as "Christian" music (e.g. notes, melodies), a better term would be "Christian" lyrics. Even in this case, however, it's difficult to define what constitutes as "Christian" lyrics. The Bible tells us that all good things come from the Father (James 1:17), so would a song written by a husband expressing his devotion and love to his wife be considered a "Christian" song? Recently, an interesting article featuring the rock band, U2, was featured in the Toronto Star. Since some of U2's songs contain "Christian" themes, does that mean these songs are "Christian" songs? Along the same lines, if you are a Christian actor, do you only find "Christian" plays in which to perform? Is it even possible to do so?
2. I think a better question is not whether a Christian singer ought to sing "Christian" lyrics, but rather "do these lyrics bring honor to God?" Psalm 19:14 says:
May the words of my mouth and the meditation of my heart be pleasing in your sight, O LORD,
my Rock and my Redeemer.
A Christian singer, after all, is a Christian FIRST and a singer SECOND, meaning honoring Christ is the most important thing.
3. This means a Christian singer does not HAVE TO sing "Christian" lyrics, but they should only sing lyrics that bring honor to Christ. How do we know whether we are doing things (e.g. singing lyrics) that honor Christ? I would suggest you ask yourself a few questions in the right order (no. 1 being the most important):
(I) What does the Bible say? This is the crucial question. We believe, the Bible is the Word of God and the final authority in all matters of life and faith. The Bible has universal application, meaning its principles hold true regardless of your circumstances, preferences, age, ethnicity or culture. This means that lying is wrong, murder is wrong, and rape, homosexuality, slander, gossip, materialism, lust and pornography are wrong! If we apply this to our example today, if a lyric gives the message that it is okay to degrade women, you shouldn't sing it or listen to it b/c the Bible tells us degrading any human being is a sin. Conversely, if a lyric gives the message that we should be thankful for every moment we're alive (such as the POD song, "Alive"), it is acceptable to sing it as a Christian (even though the song never mentions God).
4. In general, it is wise to sing and act in events that are organized by men and women
that love the Lord. I would advise only singing and acting in secular events if you have special calling to bring the message of hope of Christ to non-Christians at these events. This isn't for everyone. There are also those who are called to sing songs that honor Christ and reflect his values, but do not mention Him or address Him directly. This is also a special calling and isn't for everyone. The band POD is a good example. They are Christians, and if you listen to their song lyrics, you'll find that they sing positive songs which reflect godly values. Their lyrics do not mention Christ directly, but I think Jesus would be pleased with the majority of their lyrics.
[Answered by Pastor HM]
P.S. Media guru, Brett Ullman, came a few years ago and talked about this very subject. You can check out some of his good resources HERE.