I think bicycles are way under-rated as a bug out vehicle option. I became interested in them after reading an article a few years ago in Predator Xtreme magazine. They had a set-up for predator hunting using bicycles; had weapon mounts, tow behind cart etc. The number one benefit was covering a lot of ground and doing it quietly. There was a guy on Youtube that did a pretty cool series of videos showing the potential of bicycles. Low budget, out of the box thinking. I was literally like "holy crap, I had no idea you could do all of that".
I know for me, it's kind of a tough option. I live on the west side of Puget Sound in Washington and we have LOTS of hills. And I'm not talking gradual inclines or "bumps" in the road. I mean hills. Steep, rolling, no matter what direction you go, "in your face" hills. Which are sweet going down, but a real bitch going up, especially if you're loaded down with lots of gear.
Some options to counter that are installing one of those 2 stroke engine kits for a little help when needed, or becoming really good at getting around those hills and remembering for every hill, there is a valley (flattest ground possible). I would rather have a bicycle than be on foot that's for sure. I can create a lot of distance between me and whatever, quickly and very quietly. And as mentioned above, no need for fuel or oil. Just some spare parts, a handful of tools and a little mechanical know how.
I would recommend that if you decide to use a bicycle, find a local bike shop and take some courses on maintenance and repairs. I would also recommend taking a riding course, especially off-road. Kind of like Bicycle Cops. Those folks can do some amazing things on a bicycle. Keep the bike simple. I would personally avoid anything over a 3 Speed. I prefer a pedal brake as opposed to hand brakes, though it doesn't mean you can't install them as well. Three wheeled bikes are cool too. The kind you would see the little old lady down the street riding, with the basket. A very stable, customizeable and useable platform.
And, whether you're a guy or a girl, especially a guy, I recommend a step through or "girls" frame. It's easier to mount/dismount, and, there's nothing worse than getting "disabled" by a shot to the groin from the cross bar. I can't count how many times I've seen somebody (or done it myself) lay down a "boys" framed bike and snag the foot and trip on the cross bar as they were stepping over the bike (mostly cops), or stop, put their feet down and take one to the jewels because the ground right under them was a little lower than what the tires were sitting on.
As for me, I haven't fully committed to a bicycle yet. But I have a very large family with some younger children, so bugging out would be a hard thing to do with or without bicycles. And as we all know, cash it tight. However, I am leaning more toward at least building a "tactical" utility bike for use in my immediate and surrounding neighborhoods. Mine would have a set up for my rifle or shotgun, and at minimum, some ammo cans on the sides with medical supplies, spare ammo, flashlights, food, etc. If I do commit, I'll be taking some classes when that decision is made.
For ideas, hit Google and look up photos or articles for military motorcycles. They have some cool, utilitarian set-ups, especially during WWII. Remember, the bicycle is simply a motorcycle, without the motor. The first Harley-Davidson was born on the frame of a bicycle.