Does the holding period for RSU grants for long-term capital gains tax rates start at the issuance of the RSU or at the time the RSU vests?

The capital gains holding period doesn't start until you receive the stock on the settlement of the RSU. An RSU is just a contractual right to receive shares of stock if you satisfy the conditions to receive the shares. An RSU is not a capital asset. The shares underlying the RSUs are capital assets. But those those are not delivered until you satisfy the contractual conditions underlying the RSU, entitling you to receive shares of stock. This does not constitute legal advice and you should consult a tax advisor with respect to your individual circumstances.

The holding period for a capital asset (here stock) does not start till you own it. In the case of RSUs, the holding period starts when they get converted to stock on vesting date. At vesting date - When the RSUs get converted to stock they are income and are taxed at ordinary income rates. Held after vesting date - If held for more than one year ‘after the vesting date,’(usually happens when employee is expecting a company stock price increase) and then sold, the gain from it will be taxed at long-term capital gain rates.