Rock Band

This achievement is much easier than it sounds. Starting expert, I didn't expect to make it past the first tier of songs, but ended up completing the whole tour within a day.
First of all, don't sing, hum the songs. It's easier to stabilize your voice, it saves air, and you'll be able to hear the track better. You're probably going to fail a few times in the beginning, but don't get discouraged. Once you learn the pitch range for each song, it's easier to control your voice and your needle won't move around as much.

Some tips while you're humming the songs:
Turn your mic sensitivity to the max and if you already know the songs go to the main menu and turn off all the extra noise (instruments, crowd, etc,). This will make it easier to focus on what you need to sing.
If you're planning on playing more than 5 songs at once, have a bottle of room temperature water nearby to keep your throat from drying out.
If the line you're supposed to sing is broken up into a lot of parts or it quickly goes up and down, just hum the pitch that most of the lines line up with and don't worry about hitting each individual pitch or pausing between each word. As long as you hit most of them, you should still get a full rating.
Remember to breathe in after every measure, or whenever there is a chance.
If you find yourself running out of breath and missing notes, just pause the game and give yourself a couple minutes to catch up.
If you start failing, remember that getting an "Awesome" rating with Overdrive deployed will instantly max your crowd meter.
Don't guess at high/low notes, but instead 'slide' your voice in the right direction to hit them so you don't overshoot your mark.

Problem songs:

Reptilia

Probably the hardest song to sing in my opinion. I had to restart about 20 times before passing it. It goes from very low pitches to very high in a short period, and there are very few opportunities to catch your breath during the chorus. I would suggest using practice mode and memorize what pitches you need to hum for the high/low parts first. Making sure you have Overdrive ready before the tambourine part will make this song MUCH easier to pass. I would also recommend pausing after every verse/chorus to catch your breath. I found the hardest part wasn't hitting the notes, but rather having enough air to make it through the chorus. If you can pass this song, you can complete the whole tour.

Gimme Shelter

This song has very high notes that are hard to hit, specifically the 'rape/murder' part. My suggestion is to SING not hum these parts in a falsetto and try to have full Overdrive ready. The second hard part is the 'Its just a X away' pattern before the end, which is very hard to hit, so be sure to save some overdrive for that.

Foreplay/Longtime

In case you didn't notice, they cut off the first 3 minutes of instrumental and jump right into the vocal part, which is nice if you fail and have to restart. This song is pretty straightforward until the end. Then the singer gets a little crazy and decides to do some improve stuff with a lot of high pitches. Be sure to have full overdrive ready for this and save it until you need it. You'll know when you hit it because it tends to catch you off guard by being much more difficult than the rest of the song.

Good luck!

Just hum the songs. A lot of people get confused about the pitches and sing either too high or too low. Just practice a lot. Listening to the songs isnt as important as everyone thinks though. Once you have the pitching down it should be no problem. Good luck!