Discrete wavelet transform disadvantages
A $2$-channel stage of a wavelet transform, combines two filters in parallel, followed by a down-sampling by two. The later is the cause for shift-invariance, as the filters are time-invariant. Signals $$x_0[n] = \{\ldots,0,1,0,1,0,1,\ldots\}$$ and  $$x_1[n] = \{\ldots,1,0,1,0,1,0,\ldots\}$$ which are shifted by only one sample, yield respectively  $$y_0[n] = \{\ldots,0,0,0,0,0,0,\ldots\}$$ and $$y_1[n] = \{\ldots,1,1,1,1,1,1,\ldots\}$$

Answers to down-sampling a thus not shift-invariant. This happens even with low-pass or high-pass filters. In an $L$-level wavelet decomposition, the overall wavelet filter bank is invariant to multiples of $2^L$ shift, not the intermediate one.

When applied in 2D, classical wavelet schemes, by simplicity, apply 1 DWT on rows and columns of the image, in a separate way. The resulting 2D wavelet filter is of rank one, very poor at separating directions other than vertical or horizontal. However, genuine 2D wavelets with better  directional selectivity exist. The paper A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity, Signal Processing, 2011, is devoted to that topic.