Using ridgelet transform and Smith wavelet-like transform, new necessary conditions and sufficient conditions for a function to have uniform and pointwise Holder exponent [is proportional to] [is an element of] (0, 1) are given. We also obtain necessary conditions for function to have some Holder regularity in terms of its continuous curvelet transform. Similar to the characterization of Holder regularity by continuous wavelet transform, the conditions here are in terms of bounds of the transforms across fine scales. In 2-dimensional ridgelet transform, order of bounds in the sufficient condition and necessary condition differ by 1 in both uniform and pointwise regularity cases. Moreover, due to the parabolic scaling of the Smith transform, orders of bounds in the sufficient condition and necessary condition differ by 3/2 in both uniform and pointwise cases. However, the decay of bound of the ridgelet transform of a function with pointwise Holder regularity does not depend upon exponent. Because of the directional nature of these transforms, we are also interested in characterizing functions with directional regularity via its transform. We obtain a necessary condition for a function to have directional regularity in terms of its continuous ridgelet transform across fine scales.