  We prove that the completely irregular set is Baire generic for every
non-uniquely ergodic transitive continuous map which satisfies the shadowing
property and acts on a compact metric space without isolated points. We also
show that, under the previous assumptions, the orbit of any completely
irregular point is dense. Afterwards, we analyze the connection between
transitivity and the shadowing property, draw a few consequences of their joint
action within the family of expansive homeomorphisms, and discuss several
examples to test the scope of our results.
