SSN for H-1B individuals (Medical residents)

You can get an SSN as soon as you land.

Steps are as follows

1. Open this LINK
2. Fill in the last page with the correct details
3. Take your passport and the current I94 
4. Go to the nearest SSN office near you by checking HERE.
5. Once in the office they will take your forms and fill in the details and that's it!
6. In a week or 2 you should get your SSN in the mail!

P.S.- I do not think that the rumor that you have to wait for 2 weeks is true since I have done it 3 days post landing.