Most nations had rifle grenades. Their use varies somewhat. The Germans manufactured and issued theirs on a fairly wide basis and it saw some use particularly as an anti-tank weapon until the Panzerfaust became widespread.
The US issued 4 rifle grenade attachments per squad in their units. Most kept one or two in service. The rifle grenade was very popular with US units, particularly in Europe. The anti-tank round (the M9A1) was as accurate and effective as a bazooka with a bit less range. The troops prefered it due to not having to carry a bazooka and rockets. The standard grenade launcher used a clip to hold a Mk II grenade on it. Some units also used 60mm mortar bombs instead in urban fighting.
The British also had rifle grenades in service. Theirs were a little less popular with the troops as they could quickly damage the stock of a gun regularly used to fire them making the rifle less accurate.
The Japanese had grenade dischargers (the "knee" mortar) instead not issuing a rifle grenade as did the Italians (the 45mm Brixia mortar) and Russians (50mm mortars).