Vacuum up what you can with the nozzle, not the beater bar!! The goal is to remove as much of this soot-residue as possible. Candle wax soot for instance is a bunch of unburnt petroleum products combined with carbonized particulate matter from the wick,it has some wax content.
((Using the beater bar at this stage will spread the soot onto the shafts of the polyester fibers, polyester/nylons are petro-chemicals too, so they will BIND.))
Now you may chose between cornstarch, talcum powder, or baking soda/powder. Sprinkle liberally, combining these wont help or hurt use whatever you have or whatever is cheapest.
They will work like a sponge to some degree and help absorb this carbon-soot residue.
Now repeat the first step and nozzle off the powders that you applied.
Reapply more powder, use a clean bathtub scrub brush, or an old toothbrush, make sure its rinsed and has no bleach. Lightly comb the top from the perimeter of the stain to the middle of the stain, only brush in the inward direction, be careful not to make circular scrubbing patterns or going back and forth.
Repeat this one more time.
Now you have a few options when it comes to the liquid portion, i recommend either true soda water, make sure it contains no sugar/corn-syrup, or vinegar and baking soda/powder paste. The whole point is the effervescent fizz.
Chem-dry makes a spray that can be bought at the retail level, it has rubbing alcohol, detergents and is carbonated like soda water. Use this if your lazy, or don't otherwise feel up to following my methods, it will only remove light soot stains.
Likewise you can add rubbing alcohol or use rubbing alcohol as a stand alone step.
Alternately you can purchase all sorts of low-cost brush in carpet cleaning powders, some might already have these at home.
The last thing you want to do with soot is to get it wet prior to the dry treatment, do all other procedures before proceeding to the wet methods.
TSP-Tri-sodium-phosphate can be used at any step of the process, i would say to use it at every step. Dont use the TSP type dish-powder just because its handy, this almost always contains BLEACH!!!, Bleach is very bad for carpet do not be stupid.
Lastly shampoo the carpet. I think a steamer is better, but that means either hiring a service or a rental for most people. I think a large percentage of Americans already own a carpet shampoo machine, the type with scrubbers.
For heavy soiling you can chose to repeat all steps above.
Clear glycerin and clear liquid based makeup removers those work too, its the propylene glycol, anti-freeze would work, but most of it is green or orange, and that wouldn't be good.
MEK, Methyl Ethyl Ketone is also a very good product that is a volatile solvent which may be used to remove all sorts of soots, soils resins, and grime.
Another option is to try "white gas", naphthalene is great for cleaning circus/carnival costumes of a delicate nature from all the soot caused by the open flames of the events.
This method works extremely well if you are very careful and make sure to use small amounts keeping the open container outside or far away from occupants. It works on removing all kinds of soot. Saftey is very important, my advice is to apply a small amount of "white gas" into a small metal tin, or plastic container, then cap the naptha back up, get 20-30 cotton balls and soak them equally.
We do not want to spill naphthalene in the home. DO NOT DO THIS NEAR ANY FLAME. Make sure all candles are extinguished prior, Keep this 10 or more feet from an old pilot light stove.
Open all doors and windows in the area where you will apply the solvent, try not to hover to long in the area. It will evaporate and the smell will be gone, so don't worry.
Take those solvent dipped cotton balls and start at the perimeter of the soot stain and work inward, just like i mentioned above when using powders.
This will usually work on most any soot stain as long as the soot is not also wet with water-based solvents/cleaners.
If you already got the stain wet with a water type solvent/cleaner, blot it up, fan dry it, then use the Naptha.
Kerosene can be used too, but make sure you get a pure grade of kerosene, follow the outside preparation in the procedure protocol that i gave for the "white gas" method, this time use only q-tips not cotton balls, when using kerosene it is very important to have lots of clean white rags to remove the kerosene.
Dry ice can be used too, dry ice is a very good solvent, but not always easy to find. I clean sooty leather and saddles with dry ice, its also good at removing human oils from leather items. Petro-oils can be removed with a dry ice method from leather boots as well.
If you have the availability of dry-cleaning solvents, the type used at a dry cleaners those are good too, but hard to come by if your not in the "loop"
They have spray solvents that are usually orange scented, or even orange oils that remove tar, gum, wax, adhesive, and glue, read the label and apply to your spot, and as always start from the perimeter and work towards the center without rubbing back and forth or making circular patterns.
Some soot stains can be best removed with WAX, you will need to use white paraffin wax and have an electric iron and paper bags to remove the wax but wax is a solvent and this is why it works, im not sure exactly who would chose this method first, but it is an option to try if all else fails, because more than likely if it is that bad you will either hire a restoration service or ditch the carpet.
I remove stains all the time from expensive outfits and costumes, especially war re-enactment uniforms, opera costumes, and film stage period pieces. My background is in art restoration and textile design.
(( my worst case was restoring a white 1920s silk and cotton lace wedding dress that , was once used on set in the 70's, only to have been sprayed with hot asphalt by a street crew,it was stored in that condition for over 20 years, it later became smoke damaged with a fire in a nearby prop room, and yes i got it back to white))