I want to ask about the utility of bicycles vs. horses, but in a rather specific circumstance.
In the future, there is war (the particulars are irrelevant), and it causes the usual loss of population and infrastructure. But it's big enough to be a world-wide loss, and it happens over a series of decades. A number of people with lots of money see the collapse of human civilization coming. And they decide to work to preserve civilization with their resources (and also end up ruling the new civilization they helped save, but I'm sure that didn't figure at all into their motives).
So things collapse, but "softly". Basically, things go back to around the early 1800s in technology, then crawl up to about 1850. So, there is a steel industry of some sort, there is some manufacturing infrastructure, but there are still a lot of hand-made goods, and a lot of people are farming. But what's different from the 1800s is that the new society can take bits and pieces from modern day and keep them going, even with a weaker infrastructure.
Cars and most other ICE-powered vehicles don't exist; most of the fuel infrastructure for them just couldn't be preserved and maintained. Diesel trains become the typical long-distance vehicle.
Given that, how would people in rural areas get around locally in this society? In the actual 1800s, the vehicle of choice was a horse because... well, what else are you going to use? But in this world, there's enough infrastructure and knowledge to allow for bicycles. Not to mention that at the start of this collapse, more people probably know how to ride a bike than a horse, so they could have started with them and keep them throughout the fall.
So here's the question: what are the advantages and restrictions of a bicycle in a relatively rural environment? Assume that there is enough knowledge, materials, and infrastructure around to maintain bicycles (the local blacksmith can repair a bike much like they could shoe a horse, and vulcanized rubber is available). Would they effectively be able to go off-road, and if so, how effectively compared to horses? Could they handle wooded terrain as well as a horse, and if so, how would that influence bicycle design? Would bicycles have similar range to a horse over dirt roads? Would a local lawman prefer a bike over a horse, and would he be able to follow criminals who used horses?