It depends on your basis in the stock and how long you've held the stock. If you have had it over a year, the gains are long-term capital gains. Stock you have owned less than a year is short-term and taxed at the same rate as ordinary income.
And the amount of your social security that is taxable may increase depending on how much of the stock is taxable. The gains on the stock sale depend on the basis of the stock.
Example: If your basis in the stock is $15K, you would have a gain of 10K, and if that gain is long-term, you would still be at the 0% capital gain rate for that income. However, the $10K would increase your total income, which would make more of your social security taxable.