Nothing sucks the fun out of a victorious vintage shopping trip more quickly than a residual musty odor. Ew. Rule of thumb: if a store smells when you walk in, chances are so will any items they've stocked. Another warning sign: multiple burning candles or the scent of room fragrance. Fabrics absorb perfumes, hairspray and household odors that tend to stay with a garment long after cleaning (same goes for garments that were never worn).
Garments that can be washed are a little easier to deal with. Many swear by Borax, Wisk or Oxy Clean for major odors. Even adding a little white vinegar to the wash can help. I like Mrs. Meyers Lemon Verbena laundry detergent for light odors. Just be careful and take it one step at a time. Don't double-up treatments... you wouldn't want to damage delicate fabric.
To treat more delicate items that still smell after dry cleaning, try spritzing them with vodka or vinegar (diluted with water). Never store clothing in plastic dry cleaning bags. Remove them as soon as you get home, so they can air out and avoid the dry cleaning chemical odor.
My favorite new trick? Try fashioning a sachet of coffee grounds (sounds crazy, but it works!) For bags or shoes, place directly inside. For clothing, hang the garment inside a cloth storage bag and either hang the sachet on the hanger or place at bottom of bag. Leave overnight (or longer if necessary) and voila! Much better!