The β-γ directional correlations of several allowed β transitions with unusually large ft values (ft>7) were measured. Effects of second-forbidden β components are expected to manifest themselves in small observable anisotropies if second-forbidden matrix elements are not reduced appreciably by those nuclear-structure effects that cause the reduction of the allowed matrix elements. The experimental results for the anisotropy factor A22(W) in the β-γ directional correlation function W(θ)=1+A22(W)P2(θ) are: Na22(W0=1.69, ft=7.4) A22(W̄=1.8)=-0.001+/-0.005 Co56(W0=3.88, ft=8.6), A22(W̄=3.3)=-0.001+/-0.003 Sb124(W0=2.23, ft=7.7), A22(W̄=2.0)=+0.002+/-0.003 Cs134(W0=2.3, ft=8.9), A22(W̄=1.9)=+0.001+/-0.003 Tb160(W0=2.1, ft=8.1), A22(W̄=1.9)=+0.013+/-0.007. Except for the highly deformed Tb160, no evidence of interference effects with second-forbidden β components was found, indicating that whatever the mechanism that reduces the allowed matrix elements it somehow also reduces the second-forbidden matrix elements. The small anisotropy found in Tb160 may be a result of some K selection rule effect or may be caused by a competing forbidden β transition. Evidence for the existence of such a β transition (which, however, is very difficult to fit into the established Tb160 decay scheme) is given.