Weekly payments are certainly possible, but I challenge your claim that they are common, at least in software development. It's just a personal anecdote, but I have literally never seen it in my entire career, not in my own contract work, not when any client or employer I worked with brought on someone else, not even in a job ad on any serious contracting forum. Unless you're doing either a lot of very short-term or fixed-price jobs, I would think the overheads of sorting out invoices and tax records so often would make it too much hassle for a lot of contractors anyway, but maybe if you're used to it and have all the tools and shortcuts in place it's possible for some people.
Edit: Also, I'm talking about working directly with clients here. What goes on with agencies or other intermediaries involved might be a different question.
I tend to work with agents. It turns out a bit more expensive for the clients, but I'm garunteed to get paid, and it's often how I find out about the jobs.
Do you find direct clients from avenues other than word of mouth?
Fair enough, that probably explains our very different experiences then. As you suggested, we generally do find clients via networking and referrals one way or another.
Edit: Also, I'm talking about working directly with clients here. What goes on with agencies or other intermediaries involved might be a different question.