Attack Pattern: Buzzy Beetles attack just like Goombas in this game. They walk straight until hitting something, or falling down a pit.
Basic Fighting Strategies: You can earn some points for stomping a Buzzy, but it won't die. Invincibility will work against them.
Advanced Strategies: You can use a Buzzy shell as a weapon, but be careful. The only way to stop them is to stomp the moving shell (hard) or find a starman (even harder). The only way to kill them (without invincibility) is to hit them with a shell.

Attack Pattern: Buzzies learn a dangerous new trick in this game: they can walk on ceilings, and when Mario or Luigi passes underneath, they fall from above! Upon landing, they hide in their shell and spin in the direction they were already moving. Grounded Buzzies act just the same as normal.
Basic Fighting Strategies: You can now stop a spinnng Buzzy shell by using a tail attack (easier than stomping it). This may be particularly useful when you find Buzzies raining from the ceiling. Once again, a shell or invincibility will stop a Buzzy.
Advanced Strategies: Since you can now pick up and move Buzzy shells, they are slightly more useful (and less dangerous) than in previous games. Hammers can kill Buzzies, but they're extremely rare.

Attack Pattern: Buzzies go back to their roots in this title, no longer able to climb on ceilings. See the SMB strategies for details.
Basic Fighting Strategies: Buzzies are chumps in this game! Not only did they lose their climbing ability, but Yoshi can eat them, and Mario can spin-stomp them (you don't even need power-ups to do the latter). It's almost as bad as the fact that Goombas won't squish! Invincibility and other shells are still valid weapons.
Advanced Strategies: Eating them with Yoshi is preferable, as you can earn a coin this way. Yoshi can also kill them with a stomp. The shells are still quite dangerous, though, because timing a spin-stomp on them is no easier than a regular stomp. A cape attack, or eating them with Yoshi will work just fine.