Here's my experience when I had my Buddy's. I used 10-40 Wally World brand car oil. The cheap stuff. Got 40,000 miles of trouble free engine use out of the black 125, and it's still running strong, and 20,000 trouble free miles out of the baby blue 125 when i sold it. I changed the oil every 3000 miles, filter on every odd number.
As long as you change the oil on time what kind you use is up to you.
The difference between 10W40 and 15W40 is moot for 99.9% of scooter riders. The number before the "W" expresses the oil's viscocity (thickness) at 0 (zero) degrees Fahrenheit. If you're riding a scooter at those temperatures, you have way bigger issues than oil!
I personally bought 5 quarts of Pep Boys' house brand 10W40 when they were closing their retail operations for about $0.80/quart. It works fine.
But there's one guy on this forum who will scream that you can only get peak performance with some fancy brand of oil specifically for 4-stroke motorcycles. He should be popping in at any moment.