Crowning Music of Awesome: Several, but Do The Twist and the cover of Octopus's Garden are really quite awesome.
Ear Worm: EVERYTHING. Hear any song once, and it will be stuck in your brain forever. Even if you haven't seen or thought about the show in ten years, the moment someone brings it up, these songs will be coming back to you!
Internet Backdraft: Greg Page's return has caused some of this, especially in regards as to how Sam's departure was handled.
Older Than They Think: Most people outside Australia figure that The Wiggles are a fairly recent creation not knowing that they've been around since 1991 in their native Australia.
They've been around for so long that most Australian tropers grew up watching their videos.
Periphery Demographic: The band seems to avoid being grating to everyone but kids by having enough talent and good songs that parents and siblings don't mind being dragged along their concerts. That and they're damn catchy.
The fact they they play their own instruments (for the most part) elevates them from the normal "Triple threats who couldn't get any decent acting work" to "Musicians who specifically cater to children."
Uncanny Valley: The aforementioned Wiggle puppets, and some of their early attempts (Pre the Space Dancing DVD), to render the human members with computer graphics.