Early in the war it usually depended on the regiment or company. In some cases they were given whatever was on hand. The 2nd, 4th, 5th, and 7th Texas Mounted Rifles that served in the New Mexico Campaign went to war with a mix of civilian shotguns and hunting rifles. A number later equipped themselves with rifles picked up off the battlefield.
As the war progressed many mounted regiments were dismounted, either for lack of horses or need of infantry. Those fighting in major armies, such as the 1st and 2nd Arkansas Mounted Rifles in the AoT, were equipped with the same thing everyone else was.