I'm 30 years old, and in relatively good health. For the past week I've been having sharp stabbing/shooting pains on the sides of my breasts. More so on my right one. The pains sometimes have a burning sensation too, and I haven't done anything out of the ordinary to cause this. What could this be?

Sorry, diagnoses here is not possible. No one can diagnose random symptoms on the Internet. When was the last time you had a mammogram? I suggest a trip to your doctor and a breast exam. There are many websites showing how you can do this personally and what to look for if you don't want someone to do it for you, but a mammogram should be a priority.