As a Canadian, how can I apply for and get a U.S. credit card?

You will not be able to obtain an unsecured credit card without established credit history. You do not need a SSN to establish a credit history.

You can start by going to your bank and asking for a secured credit card. You need to begin to build your credit history; you have to prove to creditors that you will pay your bills on time. Creditors will not extend credit to someone they consider a risk. If you have no history, they don't know if you will pay your bill.

If you don't like secured credit cards, you could also take out a secure loan. Put some funds into a CD (Certificate of Deposit) at your local bank. Then, ask for a loan from that bank secured by your own CD. They will establish the loan and your payments will be reported to the credit bureaus. If you fail to pay the loan bill, they cash in on your CD, so there is no risk for the creditor. Obviously, this means you can't access your CD while the loan is active.

If you have a close friend or relative with an established credit card, you can ask to be joint owner on that credit card. All of the history for that card will flow onto your credit report.

You have to prove to creditors that you won't skip on your obligations before they will offer you unsecured credit.