Free student credit cards with bad credit or no credit history will be hard to come by in this economy. Also it will be tough to find student credit cards with instant approval right now. Most normal credit cards have phased out instant approval for now due to the economy. But with both of types, you should receive your new card 7 to 10 days after applying, or a letter within that same time frame stating why you were not approved.
However if you have good credit, or at least no bad credit history, then finding free student credit cards should be easy. However as you said, finding them with generous rewards will be more difficult. Many of the student cards on the market do not offer the best cash back. Usually it is not more than 1%.
The student credit card with rewards that I currently recommend is:Discover card for students
- 5% cashback on categories which rotate quarterly.
Please note that to be completely transparent I would like to mention that CreditCardForum does advertise Discover cards so we therefore have a material connection to the company. You can click on them to read our reviews. Also the reviews contain the regular (non-student) versions too.
Keep in mind that after a short amount of time establishing a good credit history with your student credit card, you most likely will be qualified to move up to a regular credit card with good rewards and a higher credit limit.