The only way to earn Chase Fairmont Rewards points is by using a Chase Fairmont Visa Signature Credit Card. (If you have one, the card is no longer available through the Chase site) With the card, you earn 5 points for every $1 spent on Fairmont stays, 2 points for every $1 spent on airline tickets purchased directly with airlines, at car rental agencies, and on transit and commuting. You earn 1 point for every $1 spent on all other purchases.