Peggy had a great idea with the Peterboro bicycle basket! I bought one of my own (the extra large one, in cherry), and tried it out yesterday. It has leather straps to attach it to the scoot and they fit nicely over the handlebars and fasten tightly. The basket didn't wiggle very much, and it held my sweater, purse, and the fabric I was shopping for at Walmart.
Look at the picture of this basket on their website (http://www.peterborobasket.com/p-1269-peterboro-original-extra-large-bicycle-basket.aspx ), and see where they have the leather straps attached. The only change I made was to move each strap more to the middle.
I don't think I'd want to have this basket on my scoot all the time because I'm used to being able to pull up close to things and it's harder to do that with the basket attached. I'd definitely want this basket if I was out shopping though, because whatever you put in it is easy to reach. It does make the hand brakes harder to use because it's so close to them, but the only time I use the hand brakes is if I'm going down a hill and need to slow myself down. Any other time I just let go of the throttle and have no problem stopping.
It's pretty simple to attach or remove -- just buckle or unbuckle the straps. I give it two thumbs up!