In Alas,Babylon"
Now that gasoline is scarce, two bicycle tires are worth more than an expensive car. A morality tale
http://www.novelguide.com/alas-babylon/summaries/chapter7-8
There are such things as bicycle bag and bicycle trailer. Anywhere you can drag a heavy load around with a rickety shopping cart a bicycle will be superior, especially on inclines. Post apocalypse something like what I linked to might not be readily available, but you can secure an awful lot of stuff on/from a bike rack with straps and whatnot.
Perhaps the unspecified calamity that for some obscure reason killed all living things, including insects and weeds, but spared humans, also destroyed all bicycles.
Sydney’s Daily Telegraph is somewhat obsessed with bikes and seems to satisfy the other criteria, as per this example.
Although it is not marketed as fiction.
Confirmed. Huge fan of this series, although I haven’t read the latest one yet. The gist is that Nantucket vanishes in some ‘event’ being replaced by the Nantucket of 2500 BC, and rendering high-power technologies in the rest of the world (electricity, gunpowder, etc.) useless. SCA types take over and recreate society on the 16th century French model, for the most part. Folks get around by horse, bicycle, or on foot. The drawback to bikes is likely that parts eventually run out and are difficult to make (chains, inner tubes, tires).
Sure, mountain bikes and fatbikes can be useful even on rough roads - but only until a part fails that the rider doesn’t know how to repair or replace. Also, consider:
1.) It is probably very difficult to wield a weapon from the seat of a bike;
2.) If you reach a stretch you can’t navigate on your bike, you need to push/portage it or abandon it; and
3.) It’s a lot easier to hurt yourself on a bike than on foot. I don’t refer to life-threatening injuries, or even infected wounds, but just the sorts of minor-but-painful bruises and sprains that slow you down. And those are potentially life-threatening, if they prevent you from running, or wielding a weapon effectively, or what have you. I mean, when I gouged the heck out of my left hand a couple weeks ago, I took a day off work and chilled out, and I babied that hand for a week or so. But if I needed that hand working to operate a rifle, well, I’d have had a serious problem.
4.) Related point to #3: Biking over rough terrain requires a great deal of concentration and care. And if you’re concentrating on the road, you’re not watching your rear, your flank - your situational awareness is quite poor.
Bottom line, biking will certainly be faster than walking in a lot of cases, but it confers unavoidable risks. If you don’t need the extra speed, you’re probably wise not to bother.
There’s Extinction Point by Paul Antony Jones. A mysterious red rain falls to Earth and hours afterward nearly every living thing dies. Our heroin Emily Baxter is a reporter in New York City who manages to make contact with some scientists in Alaska and she hits the road to meet them. The problem is that Emily never learned how to drive so she’s going to bike the more than 1,000 miles it’ll take to get to whatever town in Alaska the scientist are at. Oh! I almost forgot. The bodies of the dead are being used as raw materials from which alien lifeforms are spawning. Some of these alien lifeforms are extremely hazardous and attempt to kill our heroin. But, no, she doesn’t try to drive to Alaska she decides that biking it is still her best bet.
One of the main ‘good guy’ kingdoms in the books model themselves on Celts, and specialize in the longbow. They have squads of bicycle-archers that ride into a spot, set the bikes aside, shoot at the heavy troops on the other side, then ride away before their position can be overrun.
In Batman: No Man’s Land, people are seen trading bike parts for other goods.
Bicycles for military use appear to be used much like dragoons - that is, you ride the bicycle to the battle, or for scouting, but you fight on foot.
Moreover, a bike can allow an individual to carry an enormous weight of stuff over fairly rough terrain - often, walking beside the bike and basically pushing it while it is loaded with stuff. Very useful if you lack motor transport. Unlike animals, bikes don’t eat theor own loads - you get there (wherever your soldiers are going) with supplies.
Plenty on the actual real-world military use of bikes here:
Now, it is a valid point that they wear out and some of the parts may be difficult to get or make. That, I don’t know. But assuming a post-apocalyptic world, there are literally millions of bikes lying about, so for a while at least the survivors could make use of existing stock.
Future dystopia: William Gibson’s “Virtual Light” has a bike courier as a central character with some descriptions of her bike and its security system.
Alternate history: Harry Turtledove’s “In The Balance” has a scientist (I believe) make a long trip across the midwest on a bike.
Whoever it was that dismissed mountain bikes is being foolish. You can find spare parts all over, and if you get your bike from say, an abandoned bike shop, you can stock up on spare parts and spare tires when you get it. Mountain bikes would be perfect for getting around in a post-apocalyptic world, particularly if you used saddle bags in addition to what you could carry on your body.
Mountain bike parts last several thousand miles, if not abused. You’re likely to only go through tires, tubes, and the occasional spoke.
Heck, I’d be raiding bike stores just to stock up on spare parts. Something we don’t see a lot of in post-apoc fiction is survivors going to a car dealership (or bike store) to take 50 cars or bikes of the same model. This would vastly simplify the spare-parts problem and extend the useful life of the vehicles significantly.
Personally, I’d be raiding Jeep dealerships, get every diesel Liberty I could. After that, pick a good mountainbike model and grab every one of that type I could find.
Well, assuming the werewolf vampire zombie aliens don’t get me.
That comes from a scene where [spoiler]the Boss and King Arthur, traveling disguised as peasants, have been captured by slavers and sentenced to hang for killing their master. They’re in London, I think, waiting their turn at the gallows, and, while the Boss has managed to escape briefly and send a telegram to Camelot, he sees no hope of rescue because knights on horseback, even with stations for changing horses, could never cover the distance in time. But the knights do make it in the nick of time because Clarence has equipped them with bicycles.
What I didn’t get was how knights could move faster on bicycles than horses. 
[/spoiler]
I mountain bike on a regular basis and always thought it would an amazing form of transportation in a post-apocalyptic world.
First, they are extremely versatile. It would take a particularly rough piece of terrain to shut down a skilled rider and force him to dismount. And then you would just push or carry your bike over the nasty terrain or that six-car pile-up on the interstate until you were clear.
Second, a well-maintained mountain bike is virtually silent. I’m constantly amazed how often I come up behind and startle people on trails.
There would be drawbacks, though. A mountain bike that is put through its paces on a regular basis requires constant maintenance. That part is easy and I could teach someone to completely tear down and rebuild my bike in a day.
The hard part would be finding parts and tools. There are several parts on my bike, such as front fork internals, that are extremely obscure, even within the cycling world. At times they are difficult to acquire now, never mind in a world gone mad.
Same with tools. Some tools are extremely specialized and obscure and serve no other purpose in this world other than their single intended purpose such as cassette or chain removal tools. If you needed one of these tools and couldn’t raid a bike shop, your riding days would be over.
It is a poor cyclist who cannot perform a [ETA: nearly-] full overhaul at the side of the road with only what he normally carries with him. Yes, especially with a single-speed bike you sometimes have to get off and push, but the rest of the time it’s easier than walking. Finally, I present a Scotsman on a bicycle, accessorized with a Mk I MLE (Magazine Lee-Enfield). Of course you need to get off to aim your rifle, but so did horse light cavalry, the only kind left on the battlefield by 1914. For crying out loud, Germany invaded Belgium and Japan invaded China on bicyles!
I’ll see you and raise you a Scotsman on a horse.
Of course, this is why Mad Max can’t ride a bike.
This, and you wouldn’t even need a single type of bike, there are some models that are quite interchangeable parts wise. Bmx bikes are even more interchangeable although this does take you down to single speeds it means fewer parts to break.
Stirling sensibly had them organized as bike-mounted infantry. If you’ve got any road better than a wildlife path, bikes can move surprisingly fast. In Dies the Fire most of the action takes place less than 20 years after the event, so paved roads are still available for most places and even roads choked with abandoned vehicles are still passable for bikes.
Horses aren’t particularly fast over long distances. They usually gallop for a couple of miles at 25–30 miles an hour … and then they’re blowing and want to rest. They can’t sprint for hours on end any more than you can. Actually, the horse usually barely beats the running dude in the man vs. horse “marathon", which is only 22 miles. Push that distance to a full marathon, and I’d give way better odds for the human than the horse.
A rider can get a horse to run much farther than it would on its own, but at the risk of literally running it to death or at least debilitating injury. You can get maybe 8–10 miles at a gallop, or a few miles more at a combined canter-gallop for a horse in good condition before it needs to reduce speed to a walk. A walking speed for a horse is roughly walking speed for a person. Add in feeding time (horses need to graze and can live off grain and dry fodder only for short periods), slow-moving cool-down periods, watering, etc. and the average speed of a horse party might be only moderately faster than infantry.
The thing a horse can do better than a man is carry more weight (though not proportionally) at a slightly better average speed than an unencumbered man, and provide burst speed and a mobile weapons platform in attack conditions. There was a thread several years back about horseback travel, and it was pointed out that the fabled Pony Express was able to get average travel speeds of 10 mph by swapping out horses and riders at regularly spaced stations. There are events like a 100 miler (I remember reading about this when I was a horse-mad kid) where the horse has to carry a certain amount of weight (combined load and rider) over a set distance in one 24 hour period.
Contrast this with bike speeds, where recreational riders will complete centuries (100 miles) in under 12 hours. A moderately-trained bike rider can average 15 mph+ for hours (a marathon runner generally holds 12 mph+ pace) and do it day after day since they’re not pushing anywhere near as close to the edge of performance as a racer in something like the Tour de France (25 mph for 125 miles for several days). If you could switch out horses, you might be able to keep up with bike-mounted infantry, but the logistics would be a huge limiting factor. Given halfway decent roads, suitable bikes, and moderate loads, bikes would beat the shit out of horses for average speeds, and you’d have the advantage of not needing pre-supply.
If you had rough terrain, horses might be faster, but not appreciably faster than infantry on foot over the same conditions, though you could carry more in the way of supplies. ‘Course, you’d have to since hard travel would leave almost no time for grazing. Horses eat a metric assload, and that means you have to factor their fodder into your supply load. They’re also fragile, and need regular actual rest. Continual travel would reduce a horse carrying a load to a knacker’s nag in a very short time. Even leading a string of horses and trading loads on a regular basis would reduce the endurance and speed of the horses more than the endurance and speed of men on bikes.
Humans are tough motherfuckers, and we’re tuned for endurance. Give us a tool to increase our speed for an equivalent energy output, and we can not only run the prey to exhaustion, we can probably lap it in a race.