By 'classic' cavalry charges I assume you mean cavalry charging against a deployed infantry unit in the field as preformed by for example the French Cuirassiers during the Napoleonic wars? Such attacks did happen in the Civil war but not with any great success. Farnsworths charge at Gettysburg is one example I am familiar with but I am sure there were others, particularly considering the prevalency of long range raids by substantial cavalry forces in the western theater of the Civil War.

They are however an exception and for good reason. During the civil war several technological advances had made frontal cavalry charges very risky business. Charging a unit equipped with rifled muskets firing minié balls is very different from attacking a unit equipped with smoothbores. A successful cavalry charge basically relies on being able to weather the incoming fire while the cavalry closes the distance, relying largely on speed and in the case of cuirassiers, armor, to survive (ideally covering fire by cannon might also be available). This worked against units fighting with smoothbores that were accurate to 100 meters on a good day. It works less well against infantry firing smoothbores that are accurate to three or four times that distance. The cavalry simply becomes exposed to enemy fire for too long.