Unfortunately, not possible as of yet. Hopefully they'll make it so you're able to eventually.

what i do is if you have atleast one empty spot you take a random crappy car from outside and bring it in to the empty spot. ( if you have more than one spot fill them each up with a different car than the last) once there are no spots left take out the car you want to move. once out go online and buy the Elegy RH8 (must have social club profile) and replace the spot of the car you are in. You will wait outside in the car u are moving while ordering. once ordered go into the garage and replace one of the crappy cars with your car and the car you want to move will take the place of the car you chose to get rid of.

Well I sort have found a way to rearrange cars but you need one open spot in your garage and a decent amount of money.

This might sound a little weird, but its easier shown.
1) You're going to need a spot for the vehicle, put a random street car in that spot or have a vehicle you don't care if replaced. (the garage needs to be filled, but at least one spot needs to be a replaceable car)
2) Drive the vehicle you wish to move, outside of your garage.
3) Go on "Southern San Andreas Super Autos" (southernsanandreassuperautos.com) and purchase a Faggio for $5000.
4) When it asks you which vehicle you want to replace for the Faggio, select the vehicle you are currently in.
5) After you have ordered your Faggio, drive your current vehicle (one you just replaced) into your garage. Its going to ask you to replace a vehicle, and this time replace the random street car, or the car you don't have a problem replacing.

*Also, it removes the insurance on the vehicle, so thats a downside. But just buy it again before doing anything serious.*

This is hard to explain, I could make a video if you want. The glitch is a pain in the ass when you have a garage that is full of cars you want. It works much better when you have more than one spot open in your garage. But to rearrange multiple cars, you will just have to keep place swapping the vehicles back and forth.

EDIT: Didn't know a guy posted the same general idea above me. Sorry about that, started writing it before his post, but didn't finish until after.

The only way I know is blow all of your cars up and then call insurance and get them one at a time, hope that helps.