Sheevaplugs have a major problem as documented here and here and here and here... you get the point. What happens is they run fine for a while, and then all of a sudden one of the electrolytic capacitors spews its brown sludge all over everything and ceases to have correct capacitance. This is a guide on how I repaired it, and it assumes a certain capacitor is the culprit.

You'll need:

A 1000uF electrolytic capactor, rated for at least 10V, preferably 105°C temperature tolerance instead of 85°C
A soldering iron
Some desoldering wick
Rosin-core solder (lead-free is nice...)
A Hot Glue Gun and a supply of hot glue
A smallish Phillips screwdriver
Pliers

WARNING: YOU'RE WORKING WITH MAINS POWER. THERE ARE A FEW EXTREMELY POWERFUL COMPONENTS THAT CAN KILL YOU IF YOU'RE NOT CAREFUL. LET THE SHEEVAPLUG REST FOR 2 WEEKS UNPLUGGED BEFORE ATTEMPTING THIS REPAIR. FOLLOW THESE INSTRUCTIONS AT YOUR OWN RISK.

Step 1: Find All the Dead Capacitors

Apparently, the capacitors are terrible, and they eventually will explode all over themselves if under any kind of load. Look at all the electrolytic capacitors in there and see if the tops have blown out. You may have to chip off the nasty brown junk to see the tops. The only one that exploded in my case was the 1000uF.

Go through all the caps in the picture annotations.

Step 2: Desolder and Rip Out Dead Capacitors

I just yanked on the capacitor and broke the pins off, which were still soldered to the board. I used my soldering iron and desoldering wick (and some pliers) to pull the pins out. Just make sure to watch out for the tiny surface mount components on the bottom of the board.

It's hard to see the annotations below. They're on the lower-right of the board.

Step 3: Put in the New Capacitors

The picture shows the 10V 1000uF cap in shambles. I had a 50V 1000uF cap to replace it, but it was a LOT bigger. If you can find a small one that's 10V or more, go for it.

Step 4: Put It Back Together

Put it all back in the case and screw the case back together. I used hot glue to hold the power supply metal box together since my new cap was too big for it to fit together like it was supposed to. I coated the metal box cover with non-conductive tape just in case the hot glue failed and the cover came in contact with the motherboard of this little guy. I then screwed it all back together.

Step 5: Plug It Into the Wall (be Careful!)

Now that your precious is back together, plug it into the wall. This is a little scary as you've just repaired a power supply, but if you get pretty blue light and non-blinking green light, you did it! This worked like a charm for me, and only cost about 3 cents since I bought a huge collection of electrolytics from eBay a few months back.