I just finished re-watching episode 18 of season 2 of Sword Art Online, and in the end, Kirito throws Excalibur into the pit and then Shinon catches it with a spell arrow. Then, once the quest is over, he gets the sword as a gift from Urðr for saving the land. In the next episodes we don't see him use it.
According to Mansuro's answer, it does make sense why he doesn't use it that often during battles. However, it is not true that he doesn't use it at all. He did use it in the Mother's Rosario arc.
He uses it in the 21st episode. When Asuna and the Sleeping Knights are interrupted by the adventurers just before entering the boss room. Kirito appears just in the nick of time and wards them off using his own sword and the Excalibur, dual wielding them while Asuna and the Sleeping Knights head for the boss room.
He simply doesn't use it because it isn't fair, it is way too powerful. If it wasn't for Shinon, it would not have been given to him. You can think of the weapon like Mjolnir; any time a person who is not worthy tries to wield it, the weapon gets too heavy to hold.
The only time he ever used it was to fight off the guild as an equalizer. Even then, he knew he didn't have the stamina to kill the entire group.
I agree with him not using it because its too powerfull it would be unfair for others but thats assuming due to his good character.
And another thing is maybe its because he has another sword made by liz which he prefers to use better when wielding dual sword.
But I disagree with the thought of he doesn't use it much because shinon only retrieved it for him. Even if kirito left the excalibur sword and shinon doest get it back. The npc will still give it to kitaro as reward for finishing the quest, besides he was still the one that pulled out the sword.
He cannot use Excalibur as a normal sword as it is very heavy even for him. It is like in season 3, the items have a requirement to be used and probably Excalibur also has a requirement that makes it impossible to use freely, until when he used the excalibur he spent several hours sleeping.