Stains on wood can be quite persistent. This is mainly due to the fact that, by nature, wood is a relatively soft material that can take on liquid and fats easily. Wood finishes are not always one hundred percent effective at preventing stains either. Fortunately there is good news, many stains may simply be treated, this includes persistent, deeply ingrained stains.
Treat all grease stains, especially from coking oils, as soon as possible, because the longer the fat has to soak into the timber, the more difficult it is to remove. Grease stains are usually removed with water and soft soap. Use circular motions with your rags, and be sure to have a dry one nearby to soak up the greasy water that results. If you're not having luck, use a diluted mixture of water and vinegar. Even detergent can sometimes help!
If diluted with warm water immediately, alcohol should come right off wood. If left on longer, alcohol can make very persistent stains on wood. Liquid bleach or undiluted chlorine are your best bets for removing alcohol, but you have to treat the whole surface, not just the stain, because the wood will likely lighten with this treatment. Also, be careful not to get the mixture on your skin!
Candles are romantic and lovely, but their wax can create very annoying stains on wood. Ideally, you can heat the wax up a bit with a hair dryer and blot it away with a cloth. However, not all wood can be heated without warping. In this case, turpentine will work the best, but its highly flammable, so be careful.
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If you're not quite sure what made the mark, but it's colourful, your best bet is to start with warm water and soap. If that doesn't work, try melamine foam, which will remove crayon marks, shoe scuffs, and coffee spills. Be careful though, the foam will take off a bit of the wood's finish and you'll have to reapply it to protect the wood from future spills.
Wood that has been sealed with oil is vulnerable to small white bubbles, or water spots. If the spots are small, reapplication of the oil sealant will do the trick. Deeper stains might be convinced to come up with a bit of heat from a hair dryer and some rubbing with a cloth. Otherwise, sanding and re-staining by a professional is your only option.