Update: This offer is expired.
Here are some current consumer credit cards that are currently offering mega sign-up bonuses.

Barclays Miles & More Credit Card

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• Earn 20,000 award miles after your first purchase
• Earn an additional 30,000 award miles after spending $5,000 or more in purchases in the first 90 days of opening your account
• Earn 2 award miles per $1 spent on ticket purchases directly from Miles & More integrated airline partners
• Earn 1 award mile per $1 on all other purchases
• FREE Annual Companion Ticket
• 0% APR on all purchases & Balance Transfers for the first 12 months
• $79 annual fee

Barclaycard Arrival World Mastercard

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• Earn 40,000 bonus miles when you make $1,000 or more in purchases in the first 90 days after account opening. 40,000 bonus miles equates to $400 off your next trip!
• 0% intro APR on purchases for the first 12 months after account opening.
• Earn 2X miles on all purchases
• No mileage caps and no foreign transaction fees!
• Get 10% of your miles back when you redeem for travel
• Use miles for a statement credit toward any airline purchase to any destination with no restrictions and no blackout dates
• Easily redeem your miles for statement credits toward flights, cruises, car rentals, hotels and more
• Complimentary Triplt Pro subscription – the mobile travel organizer that helps make traveling easier ($49 value)
• First year free, afterwards $89 annual fee

The US Airways Premier World Mastercard

Chase Sapphire Prefered

• Earn 40,000 points after spending $3,000 within the first 3 months.
• 2x points on travel and dining, 1 point on all other purchases.
• Earn an annual 7% point bonus on all spending.
• Points are redeemable for 1% cash back or Points can be transferred at a 1:1 ratio to British Airways, United Airlines, Korean Air, Southwest Airlines, or Hyatt, Ritz-Carlton, Marriott, and Priority Club or for 1.25 cents each towards airfare.
• No Foreign transaction fees.
• No fee first year then $95.

British Airways Visa 

• Earn 50,000 bonus Avios points after spending $1,000 within the first 3 months of account opening.
• Card-members earn 2.5 Avios for every $1 spent on British Airways purchases and 1.25 Avios for every $1 spent on all other purchases.
• No foreign transaction fees.
• Every calendar year you make $30,000 in purchases on your British Airways Visa Signature card, you will earn a Travel Together Ticket good for two years when you redeem for a flight on British Airways.
• Annual fee of  $95

Marriott Rewards “Premier” Visa Signature Card

• Earn 50,000 bonus points after spending $1,000 within the first 3 months of account opening,.
• Receive an additional free night stay upon account approval, valid at category 1-4 hotels worldwide.
• Receive an annual free night certificate valid at categories 1-5 hotels worldwide.
• 5x Points on Marriot purchases, 2x points on car rentals, airline purchases and restaurants, 1 point everywhere else
• 15 night credit towards elite status.
• Every $3,000 you spend on purchases earns you one additional night credit toward Elite status.
• No foreign transaction fees.
• First year free, then $85.
• Note: You can try sending a “secure message” to chase to match your offer to 70,000 points

United MileagePlus Explorer Card 

• Earn 30,000 bonus miles after spending $1,000 within the first 3 months of account opening.
• Get 5,000 additional bonus after adding an authorized user to your account during your first 3 months as a Card-member.
• Earn 10,000 bonus miles every calendar year you spend at least $25,000 on your Card.
• Earn 2x points on United purchases, 1 point on all other purchases.
• First checked bag is free on United.
• Free Priority boarding and Non-expiring miles with card.
• Expanded Saver award availability.
• Free primary car rental CDW.
• Receive 2 annual United lounge passes.
• $0 annual fee the first year,  then $95.
• Note: You can send a “secure message” to Chase, and they will match you offer to 55,000 miles + a $50 statement credit

American Express Gold Delta SkyMiles

• Earn 25,000 bonus miles after your first purchase
• Earn an additional 20,000 bonus miles after you spend $5000 within the first 3 months of cardmembership
• First checked bag free, on all Delta flights.
• Priority Boarding on Delta flights.
• Discounted admission to Delta Sky Clubs.
• 2x points on Delta purchases, 1 points on all other purchases
• $0 intro annual fee for the first year, then $95.