Before you continue reading,
this is entirely based on my opinion. In no way are you enforced to follow this. I merely created this just to give you the brief overlook of stages to pick in tournaments or if you want a fair match. I'm not trying to control what stages you pick, you don't need to follow this. Anyways, continuing on.
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{N}eutral Stage{s} ::.
Q. What is a neutral stage?
A. As the name implies, it's a stage that's completely neutral; no walk-offs, no stage hazards, and basically the stages that people will pick for fair fights. You really might want to play on these if you want good matches. =)
Battlefield

This stage has absolutely no hazards whatsoever. With an awe-spiring scenery, Battlefield has three platforms arranged in a triangular fashion, just like in Smash 64 and Melee. The stage also has soft edges, so most characters can slide along the edges as they recover from the bottom.
Final Destination

This stage is as neutral as it gets. With a beautiful scenery, Final Destination is ideal for brawls. It's also the common pick for Smashers with a lust for fair matches. There are no platforms in this stage. The edges have lips, making it hard for it to recover overall, because your character will sometimes be stuck in the ceiling of the lip, making it near impossible to recover afterwards. Unfortunately, in playing on this stage, you are more prone to projectiles. You might not want to play here if you have a tough time dealing with spammers.
Lylat Cruise

Straight from Star Fox, you fight on the Lylat Cruise, while obviously cruising through the wonders of the Lylat System. The scenery is always changing, so you won't be stuck with a boring stage. This stage has three platforms in a hemielliptical pattern. Despite the space battles in the background, this stage will not harm you. Be warned though, while venturing through the Lylat System, the Lylat Cruise will tilt to the left or the right, depending on the scenery. Also, using Fox's, Falco's, or Wolf's down taunt will commence a cameo sequence by Star Fox and Star Wolf.
Smashville

Right from Animal Crossing, Smashville! (Falchion's favorite neutral. <3) This stage is similar to Final Destination, only shorter in length. A long, narrow platform scrolls from the left to the right. Like Battlefield, this stage has soft edges, so you can slide along during a recovery. As you're fighting, some of the cast of Animal Crossing will witness your fight. Their expressions can change as well. On Saturdays at 8:00 p.m., K.K. Slider will make an appearance, and instead of the normal songs that play, one of K.K. Slider's songs will play exclusively for you. This is by far, one of the best neutral stages to play on. **** Final Destination, does that stage have a singing dog? I didn't think so. >_>
Yoshi's Island

This stage is from the game with the same name as the stage. Yup, Yoshi's Island. One of the few grounded stages. The scenery is very well-done, everything is straight from the game. Even the seasons will change from Spring, to Summer, to Fall, to Winter, and back to Spring again. There's a really long platform at the top, which tilts as the battle progresses. Occasionally, the Support Ghost will pop up out of nowhere, saving players who are thrown off. It's a really cool stage.
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{C}ounterpick Stage{s} ::.
Q. What is a counterpick stage?
A. A counterpick stage is a stage that a Smasher will choose that is not suited well for his opponent's character. A good example of this is when the Smasher chooses Port Town Aero Drive when the opponent plays Fox. When the platform is flying through the air, the ledges can't be grabbed, therefore with a short-range Fire Fox as well as a predictable Fox Illusion, Fox will have a hard time trying to get back on the platform.
Castle Siege

Although Castle Siege is a Fire Emblem stage, Castle Siege actually doesn't exist in any Fire Emblem game. However, it's definitely themed on it. The first phase of the stage is on top of a castle. The second phase is in the king's throne room. This part of the stage has walk-offs, so be careful. The third phase is underground, balancing on a stone in a lava cavern. In between phases, you will fight on a transitional stage (which lasts a couple of seconds) as you prepare to go on to the next phase of the stage. There are also walk-offs during the transitions. Be cautious for grab whores.
Delfino Plaza

Beautiful Isle Delfino. You travel on a platform, exploring the gorgeous and nostalgic areas of Isle Delfino. Sakurai seriously outdid himself on this one, because this stage is by far one of the most well-done. Although there are tiny details changed, this is practically Isle Delfino. You play on several areas of the island, some of them having walk-offs. Be careful when you're in the water, because you're prone to a water-KO. The game's best spikers will dominate you if you're not careful. Ness is one of the easiest characters to water-KO, because his D-Air is very powerful, and it will send you straight down.
Frigate Orpheon

A ship from Metroid Prime which Samus infiltrated. The Parasite Queen can be seen in the background (hence the vagina tail o,o). There's not much to this stage. When you hear a siren, that is your cue to get into the air, because the stage will flip 180°. If you don't move, you could be stuck somewhere unpleasant. From the sides, a platform will occasionally appear from the sides. Be careful though, because the platform withdraws itself very quickly.
Halberd

Yes, Meta-Knight's sweet ride. It owns Great Fox. Anyways, you start off in a hangar, which has walk-off ledges. A platform rises, and you are to get on it, obv. If you're left behind, you're KO'd. A minute into the match, and you land on the Halberd itself. In the scenery, you can see the Combo Cannon, a boss in Kirby Super Star. This stage has some hazards, but most of them you can see them coming. If you have good dodging, you'll love to enjoy this stage. This stage also has some of the best music, imo. This is the only stage you can play from the Subspace Emissary. During the SSE, when you're in the display room (which has all the enemies), you'll find one enemy that you can't fight anywhere else; its name is Mizzo.
Jungle Japes (Melee)

Jungle Japes is friendlier on Brawl, now that you can swim. =) It's a pretty good stage. Basic, simple, fun to play on in FFAs, etc. If you fall in the water, don't panic. You still have a chance to get back on the stage, but you gotta hurry; the water's really fast. Also, don't run into KlapTraps in the water. =x Other than that, there appear to be no changes from Melee.
Luigi's Mansion

Same as Yoshi's Island, name-wise. The stage is just weird. Plain weird. Helpfully, this stage prevents most projectiles getting past the pillars (some exceptions are R.O.B.'s Robo Beam). This stage also has ceilings, preventing most opportunities for a KO. Whenever you destroy a pillar, a bunch of Boos appear for a second. When you destroy all the pillars, you get a completely neutral area. With no distractions, fight for a KO that ceilings or pillars would normally prevent. However, after a while, the entire stage respawns, sending you back to square one.
PictoChat

Didn't see this stage coming. o.O Straight from the DS, you've got PictoChat. For amounts of time, the stage proves neutral. Then obstacles are drawn onto the stage, such as a roller coaster, a whale, a giant blowing across the stage, and much more. A cool feature about this stage is that on the text screen, it reads "Now entering:", followed by a random character. You can also perform the Earthquake glitch here. One of my favorites for fun FFAs. =P
Pokémon Stadium 2

Basically a remake of Pokémon Stadium from Melee. The only difference is a graphical update, more of a view from the crowd, and you have a wider selection of field changes. Also, some of the Pokémon that appear are from the Sinnoh region. Just play here like you did in Melee. ^_^