Wasn't this what helped SMS take off originally in Europe, kids text other kids to ask them out?
Breaking up over SMS seems chicken to me; I am not especially good at conflict, but I always thought I owed it to the other party to explain myself face to face. Asking out, why not? Three hours of of somebody's evening is no huge commitment.
I guess the situation in my mind was asking a girl you already know, possibly as a friend or work/school colleague, to switch to a romantic relationship (confessing your love). There is great potential for miscommunication in those situations, as you're trying to change your respective roles. If you just met her, and so your roles are not already established, then sure, text away.
Breaking up over SMS seems chicken to me; I am not especially good at conflict, but I always thought I owed it to the other party to explain myself face to face. Asking out, why not? Three hours of of somebody's evening is no huge commitment.