Although sweat marks have staining power, you can save some tops that are treated quickly and properly. The longer a stain goes untreated, the more difficult it will be to get the stain out.
A machine washable cotton white shirt with discolored underarms is treated by using bleach with warm water. Pre-soak the area for an hour in non-chlorine or slow acting bleach (3% peroxide solution)  and laundry detergent, then wash in warm water. Try using white vinegar in your  washing machine for dark tops or tee shirts. The alcohols in deodorants can actually burn fibers, altering their color, so you will want to use a dry cleaner for silks or acetate shirts.
If you accidentally put a white top in a color wash there is still hope. Try Rit Dye Fabric Treatment Color Remover (about $2.50 at ritdye.com) or try a chlorine bleach. Red dyes are difficult to remove but you may have a better chance with blue or black dyes. Rewash the garment in hot water but do not put it in the dryer which will set the color.
Acrylic or rayon tops that can easily stretch out or lose their shape can be returned to their original form when washed and put in a high temperature dryer. Never over dry knits. Wool and cotton sweaters can be blocked by dampening the garment and laying it on a flat surface. Be sure to press with a towel to remove moisture. This is also a good way to stretch a shrunken top before it dries.
You can also save your moth-eaten top or cardigan by bringing it to a tailor or dry cleaner. A small opening, especially near a seam, can be mended but a larger hole will need to be rewoven, which can be pricey ($35-$100). A snag or pull can be easily fixed by threading the pull and pushing it through the inside of the sweater. Never cut off a snag.