The first issue, and perhaps most prevalent, is a recent WiiConnect24 update causing recurrent error codes for any and all online functionality. It only happens to a few, causing error codes 110213 and 32002 (how many errors can this thing have?). The solution? If you don't mind losing saved data (and most importantly, Virtual Console games, which are tied to a specific system), you can contact Nintendo and they'll send you a new unit within 3 to 4 business days. The other is a bit trickier; you'll have to contact Nintendo and have them send you a shipping label, with which you may send your Wii off to repair-land. This option will take at least two weeks, so it's best avoided.
Another problem seems to stem not from the Wii, but from the power supply and adapter. If you have a voltmeter handy (and you should!) you can check if the adapter if properly outputting power. If you believe this is the problem, there's a handy fix-it-yourself solution: while the Gamecube power supply doesn't quite fit into the Wii, you can make it fit. Here's some advice from reader Jay:
"I took a knife and bent out the plastic sides in order to get it to fit into the Wii. Make sure to put positive w/ positive and negative w/ negative, which is listed on the the back of the brick. But, if you're lazy, the flat (read: not round/circular) side goes in to side which has like and angled cut off piece. Well lo and behold, the red standby light turned on and the Wii worked! [By the way], the modded adapter sill works in the Gamecube, but you may have to tape it in to make sure it doesn't fall out."
Rock on, Jay. Are you guys having any other problems, and if so, do you have some possible solutions?
[Image courtesy of smashmywii.com, which, by the way, has their official video available for download.]
[Oh, and thanks Jay!]