Flexible Rewards Cards Sign-Up Bonus Offers

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American Express Membership Rewards Cards

American Express Premier Rewards Gold Card -25,000 Membership Rewards BONUS POINTS
American Express Business Gold Rewards Card-25,000 Membership Rewards BONUS POINTS
American Express Everyday Card-10,000 Membership Rewards BONUS POINTS
American Express Everyday Preferred Card-15,000 Membership Rewards BONUS POINTS
American Express Platinum Card-40,000 Membership Rewards BONUS POINTS
American Express Business Platinum Card-40,000 Membership Rewards BONUS POINTS-100,000 for a limited time
 American Express-Starwood Preferred Guest-Up to 25,000 BONUS STARPOINTS
 

I am listing Amex Starwood card as a Flexible Rewards Card above (as well as a Hotel Card) in that it offers 5,000 bonus points for every 20,000 points transferred to a variety of airline programs.

Barclays Arrival Cards

Barclaycard Arrival Plus World Elite MasterCard-40,000 BARCLAYCARD BONUS MILES50,000 for a limited time
Barclaycard Arrival World MasterCard-20,000 BARCLAYCARD BONUS MILES
  

Capital One Rewards Cards

Venture Rewards Card-40,000 Venture Rewards BONUS MILES
VentureOne Rewards Card-20,000 Venture Rewards BONUS MILES
 

Chase Ultimate Rewards Cards

Chase Sapphire Preferred Card-40,000 Chase Ultimate Rewards BONUS POINTS (50,000 for a limited time)
Chase Sapphire Reserve Card-100,000 Chase Ultimate Rewards BONUS POINTS 
Chase Freedom Card-15,000 ($150) Chase Ultimate Rewards Bonus Points (15,000  ($150) for a limited time)
Chase Freedom Unlimited15,000 ($150) Chase Ultimate Rewards Bonus Points
Chase Ink Plus-50,000 Chase Ultimate Rewards BONUS POINTS (60,000 for a limited time)
 

Citibank ThankYou Cards

Citi ThankYou Premier-currently none
Citi ThankYou Preferred-currently none
Citi ThankYou Prestige-40,000 Bonus ThankYou Points