First, it's important to define what an "injury" is for the purposes of the Vaccine Injury Compensation Program (VICP). Many vaccines have side effects, such as redness and swelling at the sight of the injection, which will not qualify for compensation. You can see many side-effects (and injuries) here.
Injuries are generally more serious than a slight, temporary fever. You may see many of the things that the VICP would consider injuries here.
This list does not include every possible injury, but only what are known as "table injuries." However, the list of table injuries includes many of the most common injuries from vaccines.
The most common injury by far is the "shoulder injury related to vaccine administration" or SIRVA. This is the most common injury because it can happen from any vaccine which has been incorrectly administered. It has almost nothing to do with the vaccine itself and everything to do with how and where the needle is placed. If someone is putting a needle into your shoulder, there is a chance that they can make a mistake. If you suffer an injury because of it, that's considered SIRVA.
However, other common injuries have more to do with the vaccines than their administration. These injuries might include things like anaphylaxis, which is an allergic reaction to a component of the vaccine, or Guillain-Barré syndrome (GBS), which is a condition characterized by rapidly evolving ascending weakness.
It is important to remember that, even though these are the most common injuries from vaccination, they are still very, very rare. For example, researchers studying the issue have calculated the rate of vaccine-caused anaphylaxis to be 1.31 cases per one million vaccine doses.
However, it is important to know that if you are injured by a vaccine, the VICP exists to compensate you. So, if you believe you have been injured by a vaccine, you should contact an attorney who is familiar with the VICP, to see if you have a potential claim.