Shellshifting is a a blanket term for several advanced techniques exclusive to Squirtle in Super Smash Bros Brawl. By exploiting quirks in Squirtle's unique dash pivot, offensive and defensive options and possibilities are enhanced, much like a wavedash in Melee.
How to shellshift
Shellshifting is a reference to Squirtle's dash pivot animation. When the player turns Squirtle around in the middle of a dash, it will retreat into its shell, slide a very small distance in his initial direction, and then very quickly slide back in the new direction over a noticeable distance before exiting its shell and resuming its normal dash. During the entire animation, Squirtle is given special physics that allow it to slide in one direction while still being actionable. By default, Squirtle will face the opposite direction of the initial dash, but pushing the control stick back to the original, pre-pivot direction will make Squirtle face that direction once the animation ends. The animation can even be interrupted by dashing before it ends. Squirtle can also stay in the shell for longer than normal by lightly pushing the control stick in the original direction. This will normally reduce momentum, but doing several pushes in quick succession will get close to the full distance.
Standard attacks and aerials
Squirtle can easily perform all its standing and jumping attacks out of a shellshift, increasing their speed and range (primarily in the eyes of the opponent, who may have previously been chasing the fleeing Squirtle). The jab, forward-tilt, and up-tilt attacks in particular are useful at interrupting the opponent should they be approaching, while short-hopped aerials like forward air and neutral airs provide explosive power options for Squirtle, allowing it to once again face its foe in the air as opposed to fighting on the ground.
Squirtle can also perform its smash attacks out of a regular shellshift, but all three of them are much better suited for use with slightly different techniques (see pump slide below, dash pivot cancel).
Quickshifting is the act of cancelling a shellshift in order to gain aerial speed. A quickshift is performed by lightly tilting the analog stick back while dashing and quickly jumping, which gives Squirtle a massive speed boost and is good for approaching with a bair or nair attack. If a quickshift is performed to perfection, the shifting sound should not even be heard, and Squirtle will get an even larger boost forward. This is an extremely effective technique against opponents who try to anticipate Squirtle's quick aerials with shielding, and also very effective for edgeguarding opponents who attempt to airdodge, as Squirtle's speed catches them off-guard when using the quickshift technique. Quickshifting can also be used to quickly short-hop over an opponent while landing close enough to execute a tilt-combo, doing as much as 50% or more damage with its u-tilt (works best against fast-fallers and heavyweights). Also, though it is extremely difficult, quickshifting can be used to do Squirtle's Hydroplane, which lessens the chance of being attacked out of a shellshift while initiating a Hydroplane, and is much more unpredictable, as one of the most noticeable times a Squirtle uses shellshift erratically is when getting ready to perform a Hydroplane. Keep in mind that in order to do a quickshift --> Hydroplane, it is 100% necessary to move from tilting back to tilting up very quickly with the analog stick. The player shouldn't return the analog stick to neutral position, but rather slide it from the back position to up position, which should make quickshift --> Hydroplane much, much easier. Also note that when done correctly, a quickshifted Hydroplane does go slightly further than a regular Hydroplane. The act of quickshifting somewhat resembles jumping as if one were closer to the actual origin to the jump.
Shellshift dancing is the act of perfectly chaining Squirtle's dash pivots together, performing each one as soon as the prior one has finished its animation. This gives Squirtle the appearance of sliding back and forth in his shell while barely exiting. This technique is useful at long-range for confusing one's foe, potentially inducing them to give away or whiff their next move, giving Squirtle the opportunity to capitalize on a possible weakness in the approach. Unfortunately, it is inferior to a proper dash dance, which has a much lower range of vulnerability and can be more easily canceled into an attack (the downside of true dash dancing is that it is far more difficult to perform). Another drawback is Brawl's tripping mechanic, which factors in with each pivot and increases Squirtle's risk of tripping mid-match, making even a regular dash dance less useful than it was in Melee.
Shellshift to grab
Squirtle's grab range can be increased greatly by grabbing while shellshifting, turning it into a high-speed, mobile standing grab, increasing the grab's range and decreasing its predictability, surprising unsuspecting opponents. This technique is superior to the standard running grab due to decreased lag and increased distance, and upon grabbing Squirtle often slides a small distance while holding its opponent, making a down-throw KO at high percents even more likely should Squirtle slide the grabbed opponent to the end of the ledge.
Hydroplane is a technique that involves combining a shellshift with a jump-canceled smash attack to slide across a tremendous distance (almost all of Final Destination) while charging a smash attack. While one of the most difficult applications of shellshifting, requiring several nearly frame-perfect actions in quick succession, the mobility and pressure rewarded for pulling it off is very much worth the effort. It can also be used to slide other moves such as his grab and tilts. It is frequently used as the first part of a Pump slide.
The Pump slide involves fully completing a shell shift (the longest possible slide the player can get) and then performing a jump-canceled up smash; essentially performing a Hydroplane. If done incorrectly, Squirtle will just slide a small distance and do its up smash (a regular dashing up-smash, or "hyphen smash"). It is easiest to do when Tap Jump is turned "on," as the game will register a canceled jump automatically upon performing the up-smash command. However, it can also be done by pressing the jump button at almost the exact same moment as A (the C-stick should not be used). Although the jump technically registers before the up smash, it should feel as though the player is pressing the buttons at the same time.
If performed correctly, several options open up for the Squirtle player. One option is to leave the smash uncharged, which will release the move at roughly the middle of Final Destination while Squirtle itself slides the rest of the distance, making for a very quick mid-to-long range KO move. Another option is to charge the smash so that Squirtle takes it with him to the end of the stage, releasing a more powerful eruption of water over a longer distance. Either of these methods can be used to simply give Squirtle some defense as it attempts to cross the stage more quickly than either its dash or its Withdraw move would allow.