By repeated conversations with your squad members on the Normandy (in-between missions), they will eventually come to you for help about a certain situation. Agree and offer to give them help, and you will be given their loyalty mission. Next time you visit the galaxy map, it will show you where to go for each loyalty mission you have (a little note will pop up, EX: “Help Thane”).
Thane’s loyalty mission is another many seem to have difficulty with. There are two key factors in failing:
If you lose sight of the Turian diplomat, you lose your lead, and you will fail.
At the end, if you don’t disarm Kolyat and stop the assassination, you will fail.
You can also assassinate the target yourself (Renegade) and still get Thane's loyalty.