Home stage
A home stage is the stage that is most commonly associated with a character, by the game and/or by players. The subject of home stages can at times be somewhat confusing in the Super Smash Bros. series. In all the Super Smash Bros. games, there are playable characters who do not have a stage belonging to their "universe". In addition, certain characters in Melee have more than one stage. As secret characters are fought on stages specific to each of those characters, and Melee's All-Star mode also gives each character a "home" stage, it's possible to determine which characters "own" which stages, even though the results may occasionally cause confusion.
Super Smash Bros.
In Super Smash Bros., each of the eight starter characters has their own stage. Of the four secret characters, two - (Luigi and Jigglypuff) - share their stages with the starter characters from their universe (Mario and Pikachu, sharing Peach's Castle and Saffron City, respectively). So they're the only unlockable characters to have a home stage (Luigi with Peach's Castle and Jigglypuff with Saffron City). The remaining two characters (Captain Falcon and Ness) have their home stages defined by where they are fought when they are unlocked. Captain Falcon is fought at Samus's Planet Zebes, while Ness is unlocked by defeating him at Kirby's Dream Land stage. Captain Falcon and Ness are considered to have these stages as their home stages (Planet Zebes for Captain Falcon and Dream Land for Ness). There is another stage from the Mario series, Mushroom Kingdom, which is sometimes considered as Luigi's stage, as well as Mario's.
Super Smash Bros. Melee
In Super Smash Bros. Melee, the situation is somewhat more complicated. While secret characters all have a stage of their own for the purpose of unlocking them, all characters have a stage associated with them for the purposes of All-Star Mode. However, for series origin purposes, Marth and Roy have no home stages, the only characters not to have one.
Super Smash Bros. Brawl
In Super Smash Bros. Brawl, stages and characters are not matched one-to-one like they are in Melee. What follows is a list of home stages based on where characters are fought in Classic Mode and All-Star Mode, almost always based on a character's universe. The only exception is |R.O.B., who has no home stage from his universe. Because of this, he has no true home stage, though as he is fought at Mario Bros in Classic Mode and All Star Mode. However no Mario characters are ever seen there in those Solo Mode games and so that stage is considered his.
*R.O.B.'s home stage is Delfino Plaza in All-Star Mode when Mario Bros. is not unlocked.
As was the case with the Past Stages in Melee, stages returning from that game are never seen in Classic or All-Star Mode. This leads to the fact that Halberd is always seen in Classic Mode, no matter what, since that is the Kirby series' only Brawl stage, and the Kirby series is always fought in Classic Mode.
Theory of "home advantage"
There is no clear or consistent benefit to a character fighting on their "home stage." Any advantages that certain characters have over others on stages happen across the entire stage list and are situational; they are not affected by the universes in which the characters originate from (for example, Fox and Pikachu dominate on Temple, even though they do not originate from the Zelda universe.) In some cases, it may be to a character's disadvantage to be fighting on his home stage; for example, Ness is unable to use his recovery move when knocked between buildings in Fourside. Additionally, Final Destination, which acts as the home stage for Roy, is frequently considered his primary counterpick, as he becomes considerably easier to combo, especially against fast-fallers. In Brawl, the ledges of the Lylat Cruise stage are thinner than most other stages, causing Fox, Falco, and most notably Wolf to curve underneath the stage more frequently when using their respective recovery moves instead of directly grabbing the edge. Additionally, in Skyworld, Pit's Centurions in his Final Smash will crash into the breakable platforms when they aim for opponents, quickly destroying the stage and temporarily removing all ledges; this consequently removes Pit's ability to plank and makes him much easier to gimp.
However, in contrast, some stages do offer advantages to some characters; as mentioned before, any such advantages are non-circumstantial, but these certain stages may have been the origination for the "home advantage" theory. For example, on the Bridge of Eldin, Zelda and Sheik's Light Arrow, Ganondorf's Beast Ganon, and Toon Link and Link's Triforce Slash are much easier to use, and harder to dodge; the frequent absence of a lower blast-line also acts as a benefit to Link, Ganondorf and Sheik, who normally have relatively poor recoveries. The low blastline of Shadow Moses Island acts as a benefit to Snake, as most of his attacks (especially his special moves) have mostly upward knockback; his high weight and falling speed make him more resistant to vertical KOs, so he himself is not as affected.
On other stages, character can receive neither a clear advantage nor disadvantage. On Summit, for instance, all the characters slide around, except for the Ice Climbers, who experience no change in their friction when walking on the stage. Like other characters, however, they are still unable to grab onto the ledges in the stage; this trait causes Belay to more frequently cause Nana to be KO'd, as while Popo could jump back onto the stage, Nana has no ledge to grab.
