Igor, actually your question is very hard and expansible even for native English speakers.
So i will add up 2 more words for further questions.
This is what a native speaker said:

"Crinkle and crumple have sounds associated with them where as wrinkle and rumple do not. Crinkle and wrinkle having smaller and more set creases, where as crumple and rumple are more loosely creased. So take a piece of paper and wad it up in your hand LOOSELY (so you have a big, light-weight paper ball that would not go far if you threw it) it would make a crumpling noise when you form the ball. Then open up the paper ball and see what it looks like....that is rumpled.
Now take a piece of paper and wad it up in your hand TIGHTLY (now you have a small, compact paper ball that you could throw pretty far) it would make a crinkling noise when you form the ball. Then open up the paper and see what it looks like that is wrinkled."

If talk about clothing, this would be an example; Girls iron their skirts when they're wrinkled (creased badly and accidentally), yet go out and buy crinkle skirts (creased on purpose). And then, when they don't like their skirt anymore, they may crumple it up and throw it in the corner of their closet.

I also searched these and understand you can use either "Wrinkling nose" or "Crinkling nose".
Again, "Wrinkling eyes" or "Crinkling eyes". With almost no difference. Even if there is a difference, it should be in their usage. So it would be nice if a native speaker mention which one of the phrases is more commonly used.
You can also google them( with these "" marks ) and compare the results numbers.

But the only difference i myself got was this=D :
For cats and dogs, we can only use "wrinkle" in this phrase:
"The cat wrinkles it's nose"
And you cannot replace wrinkle with crinkle here!


Oh i searched a lot, hope it helps you at least a little!