I would imagine in most cases you'd probably have ample time to load a vehicle pretty heavily with gear, food, weapons, etc. So, in light of that, if you really have the survival mentality, what you should probably focus more on than the specifics of what you'll grab to run with is having a reliable, tough vehicle that can haul it all with you. IMO, you could equip yourself with the best survival kit in the world, but if getting away from whatever it is that threatens you depends on the family Prius, some econobox or a sportscar, you may not be in such a good spot after all.
Think of it this way: If the vehicle is not up to a weekend camping trip down a few miles of moderate trail, it's not fit for a bug out.
The best example I can think of is Katrina. There was (more than) ample warning that the S was gonna HTF. I can understand wanting to guard your home and livelihood as long as possible, but the downside to that is you may be faced with a more challenging exit as roads become impassable by average vehicles with flooding and debris. What good is that super-dooper survival/bug-out kit if you can't get out?
I realize it's not feasible/possible for everyone due to budget, parking situation or myriad other factors, but it's my opinion that anyone serious about survival should have a simple and very robust 4x4 that is kept in good running order, whether tagged or not. What do I mean by simple? Something that you can work on and have spare parts on hand to keep it running. My personal example:
1983 Dodge Ramcharger 4x4, 4" lift, 35" tires
Carbureted 318 V8, 4 spd. manual trans, manual T-case, manual hubs
In addition to the large emergency kit and tools, I keep in/on the truck:
2 Spare tires
drive belts
radiator hoses
coolant & oil
fuel pump & hose
Alternator
voltage regulator
distributor
ignition coil
ECU
Starter/ignition relay
headlights
headlight switch
Spare U-joints
Spare front axle stub shafts
Fuses, wire, other electrical repair items
Duct tape
About $600 worth of spare parts (tires not included) on a $2,500 truck, and I'm basically carrying everything I need to keep it running short of a catastrophic failure. If I was bugging out, I'd grab batteries out of a couple of my other cars, too.
You cannot do this with a modern truck. Too many expensive, hard to replace parts that bring them down. Most vehicles made after ~1998 even have to have the powertrain modules coded to them to run, which can't be done without an internet connection and pass-though device, should the PCM get zapped somehow.
Most of us here have fantasized the TEOTWAWKI scenario, but I think how dependent we'd be on a vehicle is often overlooked. Odds of survival without one are not nearly as high; You're slow, you're exposed to the elements, and if you become injured, being on foot is extra bad.
Just some food for thought and my $0.02