An alternative to using a large wall anchor for heavy duty mounting is to always find a wall stud for whatever item you are hanging on the wall. Then you can use a smaller gauge screw (or nail) without any type of anchor required. This will leave a smaller hole in the drywall and give you all the strength you need, but limit you to the places where you can position the item.
I am not sure what the weight of the item is that you are looking to hang on the wall but another alternative could be to just use drywall screws without any wall anchor. In my house I do this a lot with light pictures, etc. A drywall screw positioned at a slight downward angle has a good amount of holding power especially if it is something that is not going to get a lot of use but just sit there (I do not recommend just a drywall screw for something like a coat hook).
And then you can always use picture hangers like this that would only make a small nail hole:
![alt text](https://i.stack.imgur.com/unVK7.jpg)
Or even those sticky Picture Hanging Strips like this:
![alt text](https://i.stack.imgur.com/UZgH8.jpg)
The Hole is too big:
It could be that your pilot holes are too big. When the screw is inserted, the anchor does not expand enough to bite into the surrounding hole. Keep in mind that most plastic anchors are tapered, so you want them to fit tightly in the hole. If you can push them in too easily, then the hole is too big. Some plastic anchors have a slight lip (collar) at the end, to prevent the anchor from slipping into the wall. When drilling the pilot hole, it must be smaller than this collar. So the anchor will not sit perfectly flush with the wall, when it's pushed into the hole.
![enter image description here](https://i.stack.imgur.com/lOYSJ.png)
Material is not stable enough:
Another common problem is that the material you are putting the anchor into, is not stable enough to hold the anchor in place. This can be particularly problematic with older plaster. You install the anchor, then when you insert the screw the expansion of the anchor crumbles the surrounding plaster. This causes the anchor to fall out, since it's essentially being held in place by dust.
Items are too heavy:
It could be that you are hanging too much weight on the anchors. Drywall and plaster are not designed to carry loads, so applying too much weight to a single point will cause the anchor to come loose.
Items are too thick"
You may also run into trouble if the item you are attaching to the wall is too thick. If the screw does not penetrate the anchor deep enough, the anchor will not expand enough to hold it in the hole.
![enter image description here](https://i.stack.imgur.com/Mr9g0.png)
Material is too hard:
You can even run into trouble if the material you are putting the anchor in is too solid. If you put the anchor in a material that is too solid (2" steel plate), when you put the screw in the anchor may actually not be able to expand. This can cause the anchor to deform and break, and again fall out of the hole.
Wrong size screw: (Thanks Joe).
If the screw is too small, it will not expand the anchor enough to bite into the wall. If the screw is too large, it can damage the anchor and/or surrounding material causing the anchor to fail.
The screw is too long:
If there is a solid surface behind the anchor, using a screw that's too long could cause the anchor to pull out of the hole as you tighten down the screw. You'll have to remove the screw and anchor, then drill the hole deeper to fix this.
![enter image description here](https://i.stack.imgur.com/sqeM9.png)
Best Answer
Plastic anchors come in different quality flavors and having the collar or not doesn't correlate with that.
I use a locally manufactured (I'm outside of US) series of anchors without a collar - their diameter is slightly smaller than that of the hole and they have small relief dots extending from the surface so that the outer diameter is slightly bigger that that of the hole. They require some force to be driven into the hole - either pushing them with a hard tool or gently hammering with a hammer. My experience with these anchors is very good - they almost never slip given the material is hard and doesn't crumble while drilling.
At the same time I saw a lot of anchors both with or without collar that slip in the hole. So I'm pretty sure it's not the collar itself that matters.
That said another very important factor is how precise the hole is drilled. It has to be of exactly the required diameter. Drilling must be performed in such manner that the drill bit doesn't exert side load onto the hole walls - otherwise the hole is milled and the anchor won't hold there.
Also usual plastic anchors can only be used in strong solid materials that don't crumble while drilling. There're special "hollows anchors" (not sure of the exact term) that can be used for slightly bigger holes and for hollow materials - they tight into a knot when inside a hollow.
Finally the anchor diameter must be properly selected depending on the screw diameter. Otherwise you're screwed.