Does cremation destroy DNA?

Or could a good lab still find some in there somewhere?

Also teeth don't always get turned into ash. It is possible to extract DNA from there as well.

Teeth and bone are ground up into a fine powder before being placed in an urn if done professionally. In Japan, there is a custom where after cremation the family of the deceased will use chopsticks to pick up bone fragments and place them in the urn (hence why it is also considered rude in Japan to pass food to others with your chopsticks).

As for whether or not it is possible to retrieve DNA, there is very little chance. Of course, you might just find a character that runs a morgue and tends to keep "souvenirs" from his/her job...

Remember, fiction does give you a bit of leeway. In your work, you can make it possible. All you have to do is come up with a reason to make it plausible for the reader. It may not pass muster with the scientists, but, hey, there's a reason it's called fiction.

I don't know if you can find DNA in ashes, but I do know that if you add such and such or do this/that to it, you can tell if the person died of arsenic poisoning. (if I remember right, it turns purple?)

Greener way of cremation...body liquefaction..is the best way

Not a direct answer to your question, but your victim might have bits of DNA that were spared cremation.

The Victorians were really into saving locks of hair for mementos of loved ones, like lockets and brooches. Or even embroidery and weird sculptures to commemorate the deceased. If they're into that sort of thing, the family might've requested hair for mementos from the body before it was cremated.

Or maybe the deceased donated hair to Locks of Love, so some cancer kid has a wig that's suddenly of vital interest to a criminal investigation. :D