A few. Rare but allergic reaction. Pain at the injection site. Muscle soreness from the injection. Possible swelling at the injection site. And if too much anesthetic give some toxicity issues. Most all of these are not too much to worry about but you did ask. It is much better to be "numbed" than to kill the handles on your dentists chair with firm grips. .
Very Few. Local anesthesia has been used since 1904! It's proven, safe, and effective. Side effects are rare. Usually, it's soreness at the injection site, or that the numbness lasts longer than expected. Rarely, there can be some nerve damage from ANY type of injection. Also, rarely you can get a elevated heart rate for a few minutes. Overdose from any medication is also possible but very unusual.
Syncope & toxicity. Syncope is simple fainting and this does occur in some patients. Toxicity reactions occur when the body is not able to break down the anaesthetic quickly enough resulting in uneasiness, shaking esp in the hands, Glutathione tabs under the tongue will treat or prevent this occurrence.