AWWWWWWWW!
I used to carry my Chihuahua with me on scoots frequently. She was actually friendly (weird for a Chi, I know), so I loved being able to take her out to pet-friendly places.
She loved being able to bark at cats and other dogs while moving at much higher speeds.
I used a front pouch carrier like
this, as well as a little pair of Doggles. She didn't like the Doggles, but she quickly realized they meant "going for a ride," so she learned to put up with them. It got to the point that I could not touch that pouch if she was in the room--she'd bug the crap out of me until I took her for a ride. I'd literally have to shut her in another room if I had to move the pouch somewhere without risk of her seeing.
I wouldn't recommend carrying your pup with you until you're pretty comfortable just riding by yourself. It's one thing to know the risks you're taking alone, but it is entirely another to do so with a pet. And don't be shocked if your dog doesn't take to riding well... some do, some don't. Just like people.
Funny story:
I once went on a week long bus trip to attend a friend's wedding. I took my dog over to stay at a friend's house while I was gone. When I got back, I parked the bike at the curb, knocked on my friend's door, and fully expected a wiggly little ball of fur to attempt to jump higher than my knees.
Nope.
Little traitor ran out the door, blew straight past me, and jumped up on the floorboard of my Scarabeo. It was very apparent that I'd created a monster.