Did you ever have a hot ash from a fireplace, pop onto your carpet? Or did someone drop a cigarette and leave a black mark on your carpet? I just found a way to fix those spots. I don't have a picture of my supplies but here are the items I used.
Shampooer (You might not need this if it's just a small spot and your carpet is already clean.)
Some kind of cleaning liquid - what ever you like. I have one called Organic Cleaner, but it's discontinued. You could use Dawn or any cleaner of your choice.
Sand paper - I used 100 grit
Old tooth brush
Below is the picture of one of the spots I wanted to remove after I vacuumed, scrubbed it with cleaner and water and used my shampooer. You might start by just scrubbing it with cleaner & water. It's totally your choice.