Although perhaps the question was directed more at finite groups: for reductive or semi-simple real Lie groups, (serious) results of Harish-Chandra (starting in the early 1950s) show that among regular semi-simple elements of the group, the supports of characters of principal series are the smallest, being confined to (maximally) split conjugacy classes, while at the opposite end the characters of discrete series have non-trivial support on all semi-simple conjugacy classes. The in-between repns, i.e., induced from discrete series on Levis, have in-between supports, in terms of split-ness.
I have the impression that suitable analogues hold for p-adic reductive groups, tho' the corresponding results are relatively much-newer (by a displacement of 15-20 years?), not to mention the complication in description of discrete series (supercuspidal repns) as induced from compact-open subgroups (Kutzko-Bushnell-et-alia). Early computations for the p-adic case go back to MacDonald (SL2) in the 1950s, and Shalika, Sally in the 1960s (SL2), Jacquet in the late 1960s, so far as I know.