The answer is no: or at least, the total angular momentum of the universe is zero. The
equations that describe our universe are predicated on the assumption that the universe is
isotropic. That means there is no preferred direction in space. If there was a total angular
momentum for our universe then since this is a vector quantity it would represent a
preferred direction. The fact that these equations accurately reflect the universe we
observe on its largest scale suggests that no such preferred direction exists.

Of course our universe isn't precisely isotropic. There are very, very small
variations in the cosmic microwave background radiation for example, but on the scale of the
overall universe these sorts of things are enough to produce an appreciable angular
momentum.
Answered by: Brent Nelson, M.A. Physics, Ph.D. Student, UC Berkeley

'If one wishes to obtain a definite answer from Nature one must attack the question from a more general and less selfish point of view.'