From the Super Mario Wiki
Spiked Goombas (known as Spiky Goombas in Paper Mario: The Thousand-Year Door, and referred to as "Spiny Goombas" by Professor Frankly) are Goombas that have thought of a creative way to avoid being squished by Mario. They wear green helmets similar to Spike Top shells to prevent Jump moves from damaging them, covering the common Goomba's weakness. They are tougher than regular Goombas and Paragoombas making the Spiky Goomba the strongest of the standard Goomba family tree. While their helmets are a great defense against jumps, they're still defenseless against most other attacks. Mario's Hammer or Bowser's fire breath are two good examples. Spiked Goombas are a staple of the Paper Mario series, being one of two variants of the normal Goomba, and have appeared the first three Paper Mario games. A similar Goomba appears in Mario Party 7 in a shop in Bowser's Enchanted Inferno!.
Spiked Goombas made their debut appearance in Paper Mario. They are first seen in Goomba Road, but also appear in many different locations, such as Pleasant Path. A single Spiked Goomba also appears at Bowser's Castle, and was in charge of guarding the items in the storage room. However, instead of guarding them, he decided to sell them to Mario and company as if the storage room was an item shop (this could be due to the fact that the Spiked Goomba had no idea who Mario is).
Like all Goombas A Spiked Goomba attacks by headbonking. Also if Mario takes damage from the Spiked Goomba's headbonk attack or if Mario jumps on a Spiked Goomba, the Spiked Goomba will laugh.
Paper Mario: The Thousand-Year Door
In Paper Mario: The Thousand-Year Door, Spiked Goombas reappear as common enemies, but they are known as Spiky Goombas. Just like in Paper Mario, jumping on it will cause Mario or a partner to take damage. Their earliest appearance is at the beginning of the game, where one battles with a normal Goomba and a Paragoomba in the Rogueport Sewers, and are later found in Petal Meadows. Once Mario obtains the Spike Shield badge, he will be able to jump on the Spiky Goombas as well as on any other 'spiked' enemy.
Super Paper Mario
In Super Paper Mario, Spiked Goombas once again appear. They, as always, cannot be stomped by normal means. However, they can be defeated by using a Pixl, such as Boomer, Carrie, Barry, or Cudge. Bowser's fire breath attack can also toast them. They appear in the Flipside Pit of 100 Trials in rooms 12 and 38.
Paper Mario: Sticker Star
Spiked Goombas do not reappear in Paper Mario: Sticker Star (being essentially replaced by Paper-Cone Goombas), but what appears to be a Spiked Goomba's helmet reappears on the Spike Helmet sticker. Spiked Goombas also remain absent in Paper Mario: Color Splash.
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