I don't think it really exists anymore, unless you'd be willing to take less pay and off the books.
I've seen it work at restaurants, bars, places like that, but normally any place now wants to count on you ANY day of the week and at any hour.
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Read the link below to see if any of these suggestions tickle your fancy:
1) Telephone Tech Support - Search around and see if there are any call centers based in your area. I used to work tech support back in college. Cable, credit card, and computer companies need their call centers staffed around the clock so there is a lot of flexiblity for scheduling. They teach you everything you need to know and you pretty much pick your own hours.. It's easy money if you don't mind being tethered to a phone for 8hrs a day.
2) Check out a site like taskrabbit.com where people post up odd jobs that they need help with. If you schedule a couple of appointments per weekend I bet you could net better than minimum wage for your time.
Try caregiving or social work. They're always hiring in my area. Some of the jobs are tough; sometimes you're just paid to cook supper and watch TV. The hiring process takes about a month, in my area. Most people that I hire give up a week or two before they're cleared to work. IDK why. Also try Home Depot. I worked at one for a while. They're scheduling policy was that you had to have open availability on the weekends, but you could set the rest of your availability. They're giant dorks, but as long as you can grin and bear it through their training and team building, the work is pretty easy and the benefits are really good. Also, try a temp agency. A lot of businesses use temp agencies for all their hiring, because the traditional application and interview process doesn't tell you crap about whether a person works or not. They pay temp agencies to fill an immediate vacancy, and if the worker is any good, they offer a permanent position within a couple months.
Can you work on cars?
I was a technician in high school, college and a few years after. A few years back I had no problems finding auto repair centers that wanted help on the weekends since they were usually busier then and were always turning away customers. Made a lot more money than a retail gig also as I could usually book 20 to 30 flat rate hours in two days.
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What about mowing lawns or painting? Odd jobs isn't so bad, at least while you're interviewing to find one you really want. Weekend bartender? There has to be something, hmmm...
I'm not sure as to where you live but you can check out Carmax if you live near one. They often hire people who do not have prior sales experience, they offer part time positions, and (from what I understand) are pretty low pressure sales environment.