As you will see momentarily, I am a certifiable home improvement moron.
How are you supposed to use these drywall anchor things?
I though you were supposed to drill a hole roughly the size of the anchor, shove it into the hole, and then screw into it. But I just did that, and when I put the screw in, the entire plastic thingy pushed through the hole into the wall! What am I doing wrong?
You’re drilling too big a hole.
You need to drill the hole just a little smaller than the diameter of the shaft portion of the anchor, so that when you put it into the wall you have to use a little force, and possibly a hammer to get it to go into the wall. The flange portion should sit on top of the wall and not go into the hole at all.
Sounds like you drilled your hole too big.
…also, hammer (or, press) it into the hole gently. Or, it will sink into the wall…putting you back where you started with too large a hole. It’s always better to drill too small and hammer /press too light…you can always step up a size. It’s a pain to undo or reduce in size.
Don’t use those. Use these or these instead. At least, if there’s going to be any load to speak of on the anchor. I am not a fan of the plastic expansion anchor.
Your suggestion is spot on, especially with the first “these” product. The screw in anchors are very easy to use, almost moron proof and do seem to support a good deal of weight.
Also, agreed with the others, you drilled your hole too large.
Thank you all.
I vaguely remember in the past trying with a smaller hole, and when I hammered it, the whole thing basically collapsed. It’s really cheap/soft plastic. I guess I need to find a happy medium. Or just use a better product, as Gorsnak recommended.
QFT! I still wish an ex-wife-thing didn’t have “my” Dremel, which is good for correcting this kind of thing.
I’m partial to this kind. Much stronger than the plastic ones, and you don’t even have to drill a hole.
You just pound them directly in? Do you take the screw out first?
(I told you I’m a moron.)
Pound them in the wall. Tighten the screw until you feel resistance (as you turn the prongs are opening behind the wall). Then unscrew the screw. Now put the screw back in with whatever you’re hanging up. Note that these types of wall anchors are permanent – you can’t take them out once they’re in. Well, you can if you’re willing to have a big hole in your wall.
I have taken them out by taking a chisel to it. With the chisel, I break off the front part that sticks flush to the (outside of the) wall. The body of the anchor falls behind the wall. There’s a hole, but not a very big one, as with any other anchor. Spackle easily covered it. 
I just put a screw partly in and tap it with a hammer to lengthen the part in the wall. Then pull on the screw and the anchor comes out enough to collapse the flange and push the remainder into the wall. Avoiding any chisel saves wall damage.
The first ones are the ones that I keep in supply for electrical work. Exit signs, lighting fixtures, conduit clips, surface mount boxes etc. They are one of the fastest and best holding. They to have a larger visible area then other anchors so if have to keep plastic ones around for those rare instances what ever is going up doesn’t cover a half inch circle.
I find the Home Depot ones to be slightly inferior to the ones sold by my electrical supplier.
All of the advice given here is spot on.
obbn, is right on the money with the screw in type, as long as you’re not trying to support a lot of weight. Personally, I like the type that gatorslap linked to.
Nice tip there, Al! 
Al Bundy
Here are my favorite kind. The anchor itself is used to bore the hole in the drywall. Then, when you put the screw/bolt in it turns a metal bar sideways. The center of The bar has threaded hole in it. As you rotate the screw/bolt in it pulls the bar up agains the back side of wall. I think this design is probably the strongest and most reliable of the bunch.
Here is an example from Lowe’s that shows the principle. I’ve seen heavier duty/better made ones also at Lowe’s but couldn’t find an example (maybe because they are called something slightly different (but they are in the drywall screw section).
These are what I’d use for hanging something heavy/valuable (or something hanging over something valuable 
)
They tell you on the packaging the exact drill size to use. That’s what you must use for it to fit correctly. I’ve seen anchor kits that even include the drill bit.
Also, watch out for wall thickness. Most home sheetrock is half inch. By anchors that are listed for half inch.
My first apartment, I screwed up one time and bought anchors for 3/4 inch. Poked it in the hole and started tightening the screw. It never got tight. Flopped up and down. The longer anchor would never work in half inch sheetrock. I had to drive the anchor all the way into the wall and patch the hole.
I like these also. I’ve had problems with the other one recommended above (the screw anchor) pulling out of the wall. Once these expand, they’re going nowhere. The downside is that you really can’t take them out and usually have to just drive them completely through into the wall cavity before patching the hole.
Sadly, in this case, there were basically no instructions. It’s a cheap wall-mounted arm mirror. It came with a packet of anchors and screws, but the instructions just showed a diagram of how the pieces fit together. No information about drill bit size or anything useful like that…
So, now that I’ve irreversibly drilled too large of a hole, is the right solution to go buy some bigger anchors that will fit properly in the hole I drilled? Obviously I could fill the holes and drill new ones elsewhere, but there’s a very small range of space in which this thing will fit correctly, so I’d really like to try to use the existing holes if I can.
You can buy bigger anchors that require a bigger hole.
I’ve done it before. 
Just like you. I drilled too big. Not a big deal. Anchors don’t cost much. The bigger ones can support more weight. So your mirror will be more secure.
I’d even suggest self-tapping molly bolts. No drilling. They have a sharp point and are designed to be gently hammered into the wall. Put the screw in and screw it down tight. That expands the molly bolt inside the wall. Remove the screw, put it in your mirror, shelf etc. and tighten.
self tapping molly bolt. Note the sharp tip. be certain you get one for half inch thick sheetrock. They come in various weight classes.
I prefer using molly bolts over anchors. Molly bolts hold more weight and are more reliable.
