Two common mistakes when addressing a carpet stain are not acting quickly enough and only blotting the stain once. A good thing to keep in mind is to treat a stain as if it was in an article of clothing — not carpet.
A red-wine stain — the mother of all stains — needs immediate attention. Blot immediately with paper towels. Repeat several times. Apply a good amount of heavy-duty kitchen cleaner and blot again. Repeat this step several times and, if results are still not forthcoming, scrub the area lightly with a toothbrush, then blot. The final step is to wet it down with cleaner one last time, lay several paper towels over the stain, then set some heavy books on the towels and leave overnight.
Grease on carpet requires a solvent, which can be rubbing alcohol, paint thinner or lighter fluid, for instance. Make sure to test the solvent in an inconspicuous place before applying to the stain (the closet is a good place). Blot with paper towels after applying solvent, but do not rub the stain. Rubbing only spreads the stain around. Repeat the process until the stain has been removed.
Cigarette burns can actually be sanded away with sandpaper. There will usually be a few black fibers, however, that will remain. Remove these fibers with scissors. If there is still a small stain or gap evident, snip a small amount of fibers from an inconspicuous area and glue them into the gap. The only person who will know is you.