I am also under 21, and have never had to mail in a paper application. I know banks have been changing up their approaches recently due to the new laws that have come out. My experiences have been as follows:
I applied for a BoA card last September (through the online application if I remember correctly), and was initially denied. However, I immediately called them and they agreed to give me an account. They asked me to verify my income over the phone and asked what the sources of the income were, but did nothing further to verify it.
When I applied for an SPG last week they said they couldn't take my application over the phone, so I submitted one online. I initially recieved the 14 days message, so I called them up. I first spoke with a general call center rep, who said he would connect me to a credit specialist. However, after a minute-long wait, the application was approved before I even spoke to the credit specialist, with no further income verification other than the information I provided on the online application.
EDIT: I do not have a co-signer on any of my credit accounts.
Starting Score: 760