Most of the celestial bodies we know are rotating or spinning on its axis.Is there any chance for the universe to rotate on its axis?
From what we observe it is extremely unlikely the universe is rotating, but it is a good question nonetheless.
Perhaps you thought that since we see galaxies moving away from us they could be being pulled outwards as a result of the rotation. All the galaxies we observe (outside the local group) are accelerating away from us, and this includes galaxies in all parts of the universe. If the universe were like a giant spinning cylinder (or whatever object you like) then the galaxies "above" and "below" us in the cylinder would appear stationary since they are rotating at the same radius.
This observation does not support the claim that the universe is spinning on some invisible axis.
As always we should remember that our local universe may not represent the entire cosmos, and we would in fact be spinning around some axis. As more observational evidence is gathered we understand more and more about the universe, and one would be foolish to rule out a theory based on current observational evidence.
As long as we are in the universe, we'll be unable to determine if the universe is rotating or not. To decide whether an object is rotating, we need some other object for reference. Since universe is a term which includes everything we know in space, we don't have a reference to watch the universe relative to, to answer this question.
I wondered the same thing until I focused my studies on the "cosmological axis of evil" regarding the CMB. It appears that some cosmologists believe that due to the fact the Cosmic Microwave Background appears to have an axis, this alludes to the expansion of the Universe having a rotation. Similar to how the earth rotates upon an axis. This, however, is not widely accepted.