The June 2009 issue of The American Organist (journal of the American Guild of Organists) ended with 7 tips John Wesley gave for congregation singing. Some of them remain incredibly salient. I'm changing the order and leaving some out.
"Sing lustily and with good courage. Beware of singing as if you were half-dead or half asleep; but lift up your voice with strength."
This is the easiest thing someone can do to make their singing better. I had a professor always say "if you're going to be wrong, be loud and wrong." Everyone knows the "joyful noise" psalm. Do it. I also don't like the excuse "I didn't know that hymn." Most hymns have at least 4 verses. If you're singing with good courage, you will doubtless know the hymn by the last verse.
"Above all, sing spiritually. Have an eye to God in every word you sing. Aim at pleasing him more than yourself."
Every religion in the word has song. God loves song. Wesley ended his advice with this. I'll end with the following:
"Sing all. See that you join with the congregation as frequently as you can. Let not a single degree of weakness or weariness hinder you. If it is a cross to you, take it up, and you will find a blessing."
Well put Mr. Wesley! If you find singing in public difficult, then you will get that much more out of it when you do it. The most important part of this advice is the word "congregation." Christian faith is built on groups of people worshipping together. Singing is a fundamental part of worship. Not participating in that part of the service means something is lacking.
Why is singing so important? We can start by looking at the roll breath plays in countless religions. Breath is an intergral part of God's creative power. Genesis 1:2states "the spirit of God moved upon the face of the waters." Breath is often symbolic of the Holy Spirit. In Genesis 2:7 God had finished creating the form of Man, but Man wasn't alive until God "breathed into his nostrils the breath of life." Breath is life giving.
So how do we get to singing from breath? The vocalized word is equally powerful. It is breath with the added power of vibration. God did not think, imagine, or will the world into existence. "God said let there be light." The vocalized word is powerful.
The most moving parts of any service are when words are said and sung. There is extra power in these words when we do it together. Don't let embarassment, fear, or lack of courage keep you from sharing in the gift of song.