When we think of bug out vehicles we usually think of trucks, SUVs, or RVs that can carry us and a lot of survival equipment. You may see highly equipped, very expensive, 4WD all terrain vehicles advertised as bug out vehicles. Most of them are way beyond what the average camper or prepper could ever afford -- or would ever need for that matter. Moreover, they may not even be very useful, depending on where you are, the nature of the disaster, and whether these humongous vehicles will be able to fit through places restricted by devastation. However, there are other options that might be more versatile. For example, I keep my dirt bikes ready to use at any time. If roads are impassable due to an earthquake or flooding, I can use my dirt bike to get to safety through or around all kinds of terrain that would stop just about any other kind of vehicle. It can also be used as a messenger vehicle during a local emergency where normal roads are impassable. If you prefer an ATV or side-by-side, it too could provide options to go places where a normal street legal vehicle can't. You' might want to find a way to attach some kind of hitch to the frame of the dirt bike or ATV. Towing a trailer will limit some of the places you can go but it will give you the ability to carry a lot more survival gear with you. You might also consider a luggage rack or saddle bags for extra carrying capacity.
An option seldom considered is a bicycle. Too often we think of them only as toys or exercise machines, but in many parts of the world they are still a primary means of transportation. Bicycles have several advantages. They are not terribly expensive (unless you go for fancy road racers or competition level mountain bikes), they require no fuel other than the rider, they are light weight so you can carry them through terrain too rough to ride through. A good bike could be your best bet for escaping an unsafe situation. You don't see too many folks pulling trailers with bicycles (except in third world countries), but it is possible and would increase your cargo capacity. If you are in average physical condition you should be able to pull up to 600 lbs on flat ground, which gives you considerably more room to bring along what you need than you'd have in even the biggest back pack. The capacity over hills and rough terrain would, of course, be less, but still much more than you could comfortably carry on your back. If you decide to include a bicycle in your survival gear, the best bet would be a mountain bike, with off road tires and multiple gears. Racing bikes are great on pavement, but the narrow tires wouldn't handle damaged roads or going off road if necessary very well. A cruiser style bike will have tires capable of handling more types of terrain but usually are only one speed so they wouldn't be as good for negotiating hills or heavy loads. As with a dirt bike, you will probably have to engineer your own trailer hitch if you do decide to pull a small trailer or you might want to add a luggage rack or saddle bags. The little baby carriers you see towed behind bicycles might make a good starting place for bicycle hitches and trailers. Bicycle baskets that attach to the handle bars can carry some things.
You will need a pretty small trailer to use behind a dirt bike or bicycle. There are some trailers designed to pull behind ATVs and some for street bikes that might be a starting point. Even a little garden cart designed to be used with riding mowers and garden tractors might work, but they usually aren't designed to be particularly light weight. You might also consider building your own trailer from scratch. The trailer kits you can find at places like Harbor Freight will usually be too big and too heavy for pulling behind a bike (pedal power or motorized). You might make one from bicycle or tricycle parts or even use a child's "Radio Flyer" style wagon. Some of these are available with big, soft, all-terrain tires designed for garden use and would adapt well for off road towing.
You will need to limit the size of the trailer for several reasons. First up, you don't want it to be too heavy. The lighter the trailer, the more gear you can carry and the easier it will be to pull. Secondly, you want something that you can pull through small spaces without difficulty. In really difficult terrain you might even have to pull the trailer by hand or even be able to lift it over obstacles. You will also want to make sure it is sturdy, dependable, and easy to maintain.
I have seen neat little tent trailers to pull behind ATVs that would even provide shelter, but you will sacrifice some cargo capacity if you choose one of these and they would probably be too big and heavy to reasonable tow behind a bicycle, but you might make it work behind a motorcycle. If you need or want to be able to camp along the way, an easy, light weight, and inexpensive alternative is to toss in a small tent or maybe just a tarp you could toss over you and the trailer to protect you from hot sun or bad weather.
In a pinch you could rig a travois behind a bicycle or dirt bike to carry your gear or even transport an injured person. You need a couple of poles lashed together at one end to attach to your bike with a tarp, blanket, or net stretched between the other two, spread out, ends to carry your load. The advantages of a travois are primarily the ease of construction from readily available materials and the lack of moving parts that could fail. The disadvantages include increased resistance in dragging them and fairly rapid wear on the trailing ends. But they might be a very good option in an emergency for transporting a load or injured person a short distance.
Speaking of bicycles, I have even seen some three-wheeled rigs with a camper attached, looking kind of like kind of pedal-powered Class C motorhome. While those might provide the ultimate comfort for bugging out, do their height , width, and weight, they wouldn't be very practical on damaged roads or rough terrain and I, for one, would not want to try to pedal one of those up much of a hill! Likewise, there are custom made motorcycle based Class C motorhomes that would provide many of the features of a larger motorhome as a Disaster Recovery Vehicle, but might have greater range due to lower fuel consumption and, their smaller size might let them go places larger vehicles won't fit.
Here is a slick Youtube video of 15 Minicampers and Motorcylce Campers that might be useful for recreational outings as well as for an emergency bug-out. Many of them include quite a bit of cargo capacity for carrying survival essentials.
One of the chief characteristics of survivalists is their ability to innovate. Be innovative in your preparations as well as in your survival techniques.
Think outside the box!