Stone-Eye

Stone-Eyes are giant gray stone heads that debuted in New Super Mario Bros. U. They appear to be based off of the s of, and also bear a passing resemblance to Thwomps.

New Super Mario Bros. U / New Super Luigi / New Super Mario Bros. U Deluxe
In New Super Mario Bros. U, New Super Luigi U and New Super Mario Bros. U Deluxe, they mostly inhabit desert-themed levels, and either move in numerous different patterns (e.g. swaying, up and down, to the left or right, etc.) or are completely stationary. Based on their movement patterns, they can assist and/or hinder players. A small Stone-Eye can be seen in the Records Toad House on Secret Island.

Some Stone-Eyes can be moved by Ground Pounding on top of them; this is often necessary to reach things such as Star Coins or paths to secret exits. They have their eyes fixed on the player when approached; if the player performs a Spin Jump, their eyes will roll around for a moment.

There are also brown-colored, rectangular Stone-Eyes with their eyes closed that serve as stationary platforms. In New Super Luigi U in the level Spinning Sandstones, one of them resembles Luigi.

Super Mario Run
Stone-Eyes reappear in the background of desert levels in Super Mario Run.

Super Mario Maker 2
In Super Mario Maker 2, Stone-Eyes can be seen again in the background of New Super Mario Bros. U-styled desert courses. As of version 3.0.0, Stone-Eyes designed in the style of Super Mario World may appear on desert-themed course icons in Super Worlds.

Mario Golf: World Tour
In Mario Golf: World Tour, Stone-Eyes appear on the Layer-Cake Desert golf course. The course features reddish totem-like Stone-Eyes that act as obstacles, as well as regular Stone-Eyes that are out of bounds.

Mario Golf: Super Rush
In Mario Golf: Super Rush, Stone-Eyes appear in Balmy Dunes. In Golf Adventure, there is a mission where the player needs to shoot golf balls into a pot placed on top of a Stone-Eye's head.

Mario Party 10
Stone-Eyes appear Mario Party 10 in the Fruit Scoot Scurry minigame. They serve as platforms that players can jump between to collect fruit.

Dr. Mario World
Stone-Eyes appear on the map of World 2 in Dr. Mario World once it is completed.

Trivia

 * Prior to their appearance in New Super Mario Bros. U, Stone-Eyes made an appearance in the New Super Mario Bros. Mii tech demo.