“Your horoscope for today: Today is a good day to be trapped on a frictionless surface. Lucky numbers are 12, 13, 22, and 36.”
I’d be more concerned with air currents. I’m pretty sure other planets would have a negligible affect when compared to possible local effects.
Or indeed, the way jellyfish move.
Aren’t we forgetting gravity? Surely the surface would have to be curved according to the local gravity field? Otherwise gravity makes you slide off in the appropriate direction.
A perfectly flat frictionless surface on a gravitationally-perfect sphere is a nice trap, because gravity attracts you to the middle of the plane and the middle of the sphere and the middle of theplane is the closest to the middle of the sphere.
If the surface you are on isn’t perfectly rigid, you could propel yourself off of it by running or crawling. You would mostly slip, but your weight deforms the surface and you send vibrations outward through the hemisphere of whatever the surface is made of. If you are putting your feet down to the north, slipping them southwards and picking them up to the south, the vibrations you transmit are asymmetric with respect to latitude, and they convey momentum southward. Your body mass has to translate northward to conserve the total momentum of the system.
This is a little taste of what swimming in mercury would be like. When you are swimming, you are deeply immersed in the liquid, and on our frictionless surface you are only sinking in a tiny bit, but in either case you are essentially using the inviscid, inertial behavior of the medium. As long as the deformation you give the surface is nonzero, you can use this method.
So let me see if I’m understanding how this would work. We’ll symplify the problem by assuming you’re in a spacesuit with no detachable parts in a vacuum, so the only movement can be caused by the motion of your body and the contact with the frictionless surface.
Now if you start motionless standing up straight and facing north, your feet are at a point we’ll arbitrarily call zero and your center of mass is directly above this point zero. Now if you bend your knees and fall backward, your center of mass will no longer be above your knees. Based on what people have said, would this mean that your center of mass would remain at point zero and your feet would slide to the north?
Yes.
BTW does the suit have a valve on the air supply that you can use for reaction thrust? If not, can you make a break or tear in it?
My applied maths teacher would’ve strolled through this problem and left me feeling a bit thick for not having seen each of the small steps necessary to solve the problem. However, as a) it’s 18 years since I sat applied maths and b) Colonel Robinson is almost certainly dead, I have no idea how to solve this now.
Anyone care to take a stab at it?
I’m pleased to hear that farting would work. 
How about a water pistol?
Or a fire extinguisher?
Anything that moves mass away from you will work. See above discussion about conservation of momentum. Also, you can think of you and everything attached to you as a system with a center of mass. If you throw anything away, the center of mass moves just a little closer to that object, and you will move in the opposite direction enough to keep the center of mass in the same place relative to the surface.
I suppose peeing is an option as well.
Vectored thrust. 
I don’t remember all the details, but start with a force diagram. There is a normal force upward from the floor, a normal force sideways from the wall, and of course the weight of the ladder, which (effectively) acts at the center of mass. The weight creates zero torque, but the normal forces do not. At some point, the wall normal force should drop to zero. The question is, when exactly does that happen? Keep in mind that if the floor end of the ladder is x units away from the wall, and the wall end is y units away from the floor, x^2+y^2=L^2 if the length of the ladder is L. This might be enough to get you started.