Length contraction is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer. Length contraction arises due to the fact that the speed of light in a vacuum is constant in any frame of reference.

Has length contraction been observed?

Length contraction has never been directly measured. ... Current-carrying wires aside, there is little doubt that Lorentz contraction occurs if length is measured – that is, when different points on a moving object are all measured at the same instant of time in the observer's stationary frame of reference.

Why there is no contraction along the perpendicular direction?

Length doesn't contract in the direction perpendicular to the velocity, since there is no motion relative to the rest frame. I think you misunderstood something. Length contraction in a moving body,does not occur in the perpendicular direction of its velocity (not length as you mentioned).