I asked this very question from Keswigs when I bought my wig from them a while back. Turns out, it depends on the material of the wig. If the wig is made of synthetic material, you can't style or color it; but if it's made of natural human hair, you can. So guess what I bought? Natural human hair wig, of course!
I didn't want to buy many different wigs for different occasions. Of course, as women, we want to style our hair differently every now and then, and we also want to experiment with colors, but because I didn't want to spend too much on buying 3 to 5 different kinds of wigs, I decided to buy one wig made from 100% natural human hair.
I can curl it, straighten it, and I even tried to dye it once but realized hair colouring should only be left to professionals. But just like our natural hair, styling and coloring our wigs can lead to damage so their lifespan will be cut short. For my natural human hair wig, I used it for about 7 months before I replaced it, 'cause one thing, I got bored with it, and the other, I started to notice some wear and tear.
Now you might be wondering why synthetic wigs can't be styled or dyed. Well, they're usually made of polyester or acrylic, which are difficult to dye. They're made in the lab and contain chemicals that give them a permanent feature - if your wig is straight, it will remain straight, if it's curly, it will remain curly.
While synthetic wigs are much cheaper, they're also not versatile so you'll be stuck with one color and one style. Unlike natural human hair wigs, though a tad more expensive, can be styled, can be colored, so they're versatile to use.