All chat domains, all rooms: this is fine
Typing the message this is fine as-is results in server side rendering of the message into this:

This isn't case sensitive, however formatted messages won't be parsed as a picture.
Meta Stack Exchange/(any room): red tags
Send a message consisting solely of [status-SOME_PLAINTEXT_STRING]
and it will appear in chat like a moderator tag. Doing this as a reply to another message is okay, but other text or formatting will not work. The plaintext string cannot be longer than 50 characters, and must match the regex [-A-Za-z0-9_]+
.
Gaming/The Bridge: minigame Eggs.Asteroids
Send the message insert coin
and click on the floating "click me" button that appears to start an Asteroids-esque minigame with user avatars.
Super User/Root Access: helpful assistant Eggs.Assistant

Send a message matching the regex /(?:^|[.!?:]\s+)(?:(?:how\s+(?:can|do)\s+i)\s+([^?!.:]+)\?|(?:i(?:\s+want|(?:\s+am|'m)\s+(?:wanting|trying)|'d\s+like|\s+would\s+like)\s+to\s+([^?!.:]+)(?:$|\.|!)))/i
to start an animation of a "helpful" anthropomorphic paperclip. Some other "how" questions will also work. (Yes, it's kinda crazy, but that's what balpha says the official regex is.)
Server Fault/The Comms Room: system crash Eggs.Console
Send the message rm -rf /
to blank your screen out for a few seconds.
Stack Overflow/(any room): insanity Eggs.Cthulu
Send the message <(?:"[^"]*"['"]*|'[^']*'['"]*|[^'">])+>
to make quotes from this famous answer float all over your screen. Some other regular expressions will also work; the system looks at each attempt and makes a guess about whether it's an attempt to parse HTML or not.
RPG Stackexchange: dice roller
Send a message with d4, d6, d8, d10, d12 or d20 to have chat roll a die matching your input. This can be preceded with a number from 1-9. Chat will return a set of die rolls matching the dice expression used. You can also mix the expressions to get die of multiple types in a roll.
Travel Stack Exchange/(any room)
The world map preview on the bottom bar to the right of the message entry box zooms in and out based on how far the page is scrolled: scrolling up will zoom out, and scrolling down will zoom it in. The center point of this zoom changes based on your location.
Other SE sites/(any room): search
Searching for "easter", "egg", "eggs", "easter egg" or "easter eggs" results in a number of fake messages instead of actually searching for those terms. The results change each time the search is run. This works on all the chat sites
Print Scrn
Eggs.*
easter eggs, or it isn't present on any of the sites I checked. If I had to guess, I'd say it's available only in secret moderator-only chatrooms. The Wheel of Blame is far too powerful for the likes of us normal community members.