I've excluded pickups and most SUV's from this discussion for one reason only.
Lack of fuel available will make their use restricted to the initial bugging out trip.
There is also the very real possibility that larger vehicles may not be able to even do that.
If Hurricane Irma taught those of us in Florida one thing, it's don't think gas or food
will be available on your bug out route.
I-75 backed up, cars out of fuel, gas stations closed (out of gas) food at rest stops gone, etc etc.
And this was before the Hurricane even hit.
Many spent the worst part of the storm sitting in their cars at a truck stop.
All this trouble and this was during an orderly and advanced notice evacuation.
Imagine the chaos with no warning?
Oh, and for those who say they have plenty of gas,
tell that to the people stuck in traffic jams on I-75, I-4, and I-95. It won't be pretty folks.
Those who live or work in the city are going to find getting out is not going to be easy.

I'll just list the various methods people have talked about and the plus's and minus's of each.
Walking, bicycles, motorcycles, golf carts, and atv's will be invaluable in a crisis situation.

Walking:
A pretty obvious transportation mode. Basically has the benefit of you not needing anything
other than healthy and strong legs.
Very stealthy way to get around. Quiet and with many people in the same boat,
you're not going to stand out.
You obviously don't have anything to steal so not much of a target.
Basically you can hide at any moment and easily.

Walking Downside:
Your health is number one. Few people actually walk much and then it's in the best of conditions.
In addition, you're probably not carrying a back pack of any size.
While there are some who back pack on a regular basis, the majority of people don't.
Hence the amount of food or gear you can take with you is going to be very limited.
The amount of distance you can travel will also be limited for most people.
Small children will become a huge burden to deal with slowing your travel
and adding to the items you must bring with you.
You only have to watch parents with small children in an airport to know this is going to be tough.

Bicycles:
A huge step up from walking in many ways.
The amount of distance you can actually travel is much greater.
With a rack on front or back, you can carry much more gear or in a back pack with less effort.
Cyclocross, cross bikes, or Hybrid bikes have slightly larger tires and can go thru some
pretty rough areas with their more agressive tread design.
Equipped correctly, they could also transport small children with you.
Like walking, easy to hide somewhere taking the bike with you.
Added option of having a small pull behind trailer for your bike.
A possible cure for the small children issue.

Bicycles Downside:
If all you own is a 10 speed road bike, you might as well walk.
In rough terrain or debris strewn streets, you WILL cut a tire.
Now you're walking and dragging a broken bike. Make sense to you?
There are not as many options for carrying cargo or small passengers with a road bike.
You become a slightly larger target of people who want your bike.
While an all terrain bicycle is a good idea, the effort expending will be more than a hybrid
hence limiting your range but a good option none the less.

Motorcycles:
I'm not talking about a Harley here. Most guzzle more fuel than some cars.
I'm talking about either Off road or Enduro type bikes.
Something on the order of 100cc or 125cc in size. Many get 100+ mpg.
Somewhat limited in speed to around 55mph but more than enough range to get you quite far.
Even something in the 250cc range is good on gas. Better choice if there are two of you.
With racks they have the ability to carry more gear or a passenger or both.
Like all-terrain bicycles, you have the ability to go where there are no roads or to navigate around
obstacles in your way on the road.
Yes, you do become more of a target for the bad guys but at least you can outrun them.
This is perhaps my top choice for someone who works in the city far from home.
Rent a storage locker near to your work and keep the motorcycle there
with a couple of extra gallons of fuel and a small bug out bag to get you home.
Most motorcycles of this type will easily get you 2 hours of drive time on a single tank.

Motorcycle Downsides:
They do depend on gas. AND the gas needs to be treated when you're not using it.
They need to be maintained in running order. You can't just forget them.
They make noise. Not the most stealthy way to get around.
You will become a target for thieves. If you have one, keep it hidden at all times.

Golf Carts:
While some might find golf carts not to be a viable option, they can be.
They have huge cargo carrying capacity and can take more than just one passenger.
While not a bug out vehicle, a golf cart can be very handy around where you live
or at your bug out location.
Even such simple tasks as moving firewood is much easier with a golf cart.
For local, around your area, they can be a good option.

Golf Cart Downsides:
Somewhat limited in range is the big one. Gas ones, not an issue with spare fuel.
Gas ones, however, are very bad on fuel mileage but locally who cares.
Electric golf carts are a different story. You are limited by the charge in your batteries.
That said, golf carts routinely run around 18 hole golf courses on a single charge.
However, that is also another downside. You need a way to charge them.
Another downside is the terrain can't be too rough. Not a, through the woods, vehicle at all.

ATV's (All Terrian Vehicles):
Not a vehicle I'd want in the city at all. BUT, take it out to the country and it's almost perfect.
Small enough to go anywhere in the woods and powerful enough to take
a lot of gear and a passenger easily.
The number of ways I use mine around my home is endless.
On trail rides in Virginia I was riding the entire day on one 4 gallon tank of gas.
I don't think there is a farmer alive that doesn't own one. They are just that versatile.

ATV Downside:
They need gas.
Although I find mine very good on gas, I still need to store some along with the generator gas .
They are noisy. If you own one, keep it hidden as someone will try to steal it.
The plus side is there is little chance they could find you or catch up to you.

In Conclusion:
So what's the final verdict on what to own?
If I could only own one vehicle for shtf or disaster, it would be a small dirt bike.
Uses very little gas and can, literally, go anywhere.
Easy to store and hide away even in the city.
Fast enough to get you out of harms way.
Powerful enough to even get two people on the road with some gear.
2 people, go with a 250 cc bike.
Most will have a 200 mile range on a single tank of gas.

I feel I have to add a personal comment on this subject which might pertain to others.
I know someone who has a great bug out location and well stocked. Actually a 2nd home.
However, they live in a big city and their position is
they will stick around to protect their home as long as possible.
This, in my opinion, is a huge mistake.
While this might be the correct path to take in a normal power out situation or Hurricane,
it is a recipe for disaster if it turns out to be "The Big One".
People are going to know very quickly if this is not normal and will panic almost immediately.
It's very possible there will be no way for them to bug out or move or even get home.
As in my thoughts on Hurricane Irma, if people can't get anywhere with advance notice,
what's it going to be like when SHTF? You're not going anywhere.
It's my position that it's better to overreact, get out and
admit you overreacted, than wait too long.
A plan is worthless if you don't have one and worse if you don't follow it.