I'm pretty sure it's NWA, Midwest, and Delta that have the most pet friendly policies:
http://www.tripswithpets.com/petpolicies.asp
a LOT of airlines (most) will not allow ANY pets in the cabin, some not at all, and most will not allow anything like rats (they stick to cats and dogs).
Of the ones who do allow animals as carry on, the limit is usually 1 animal, sometimes 2, but I don't think any will take 3. You may be able to do some persuading to let 3 rats stay in a carrier, but you would be hard pressed to get more than that. Also keep in mind that with the stigma against rats, and that airline personnel can still turn your animals away at
check in, you will have extra difficulty even getting them onto the plane.
You can call in, but don't specifically ask about rats - people will tell you that they are not allowed, except in the airline policies online, they don't specify rats - people just guess because they don't know their own policies, making up answers as they go. Many small pets are allowed, however most airlines wont take them as carry on, and people are just personally afraid or grossed out by rats. If you call the airline, say 'small pet' or 'small rodent' and specify that it is impossible for them to get out of the crate permitted for the particular airline.