In Super Smash Bros. Melee
The Archives, found in the Data submenu, allows players to view two videos:
Archives are only available in the NTSC versions of Melee; the PAL versions remove the menu and its contents. It is assumed this was done due to the presence of more languages being available causing space concerns.
 In Super Smash Bros. Brawl
This area, found in the Data sub-menu as well, allows a player to watch a video from the following options:
 In Super Smash Bros. 4
It is confirmed that the fourth game will feature "introduction" clips for various characters, relating to an Adventure Mode equivalent.