Hello, Sarah --
Your capped bust half dollar is a valuable coin, but it sounds like wear and damage has lowered the value of your specimen. Figure that a dealer might pay you $25 US dollars for your coin, and then sell it to someone else for about twice that much. Without the stains and spots you describe, the value would be about twice as much.
NEVER CLEAN A COIN. CLEANING RUINS VALUE.
The US bust half series is an interesting one. Just about all the coins are valuable. The common kinds, like Sarah's 1931 example, catalog around $50 to $100 for a worn, but problem-free specimen. As condition improves, so does value. An almost uncirculated coin can approach the $500 mark, even for common-date coins. Here is a summary for most dates of Capped Bust halves. Coins must be free of problems like scratches, stains, spots, nicks, gouges, and cleanings to qualify for these values:
worn: $70 US dollars approximate catalog value
average circulated: $130
well preserved: $350
fully uncirculated: $420
The coin in our picture comes from CoinZip.com, a web site full of great info on US coins. The CoinZip specimen falls somewhere between 'average circulated' and 'well preserved' giving a catalog value around $200 to $250. It is a nice coin, and CoinQuest thanks CoinZip for use of their coin photo.
There are some special dates which qualify for higher values. Here are the special dates with values for average circulated coins:
SPECIAL DATES AND VARIETIES:
1807, bearded Liberty: $1250
1812, 2 over 1, large 8: $10,000
1812, single leaf below wing: $2000
1817, 7 over 4: $100,000
1820, no serifs on Es: $500
1830, large letters: $4500
1836: $1250 (must have 50 CENTS spelled out on eagle side)
The better dates are slightly uncommon, with values ranging from 1.5 to 3 times those of the common dates:
BETTER DATES AND VARIETIES
1814, 4 over 3
1817, 7 over 3
1818, 8 over 7
1820, 20 over 19
1822, last 2 over 1
1823 broken 3 and patched 3
As you can see from the list above, there are many subtle varieties of these coins that collectors follow. The varieties are too hard to explain on this forum, but if you are interested, seek out a knowledgeable collector, buy a book on coins, or join a coin club.