The bicycle is the best way to travel and so it is reasonable to assume you will want to carry your ukulele on your bike. It is always a good idea to carry your ukulele in a good hardcase, my favourites are the Gator case for soprano ukuleles and Ohana cases for all other sizes. Then you will need to somehow attach the ukulele to your bicycle.
Jim of the Night Owls always carrys his soprano ukes in a back pack and indeed this is an easy option for the smaller uke.
I have always used a Carradice Camper Longflap saddle bag, soprano or concert ukuleles fit neatly under the flap.
When cycling to work a few years ago my ukulele slipped out from under the flap and narrowly missed the wheels of a passing bus! It is important that the case is secured by tying the side flaps through the handle of the case and so preventing it slipping out from under the main flap. I have shown this in the picture below…
MoolyMoo (Mrs Nipper) uses a pannier and bungy cord combination when carrying her ukulele by bicycle. I find this method is also useful when carrying the bigger sizes of ukulele.