Disadvantages of wavelet transform
If you consider the whole set of potential wavelet transforms, then you have a lot of flexibility. 

For instance, should you use 1D continuous complex wavelet transforms, by analyzing the modulus and the phase of the scalogram, and provided you use well-chosen wavelets (potentially different for the analysis and the synthesis), and a proper discretization, you can:


find discontinuities and irregularities of a signal and its derivatives 
find break point location by wavelet ridge extrapolation
denoise
perform matched filtering based on templates (with complex continuous or discrete dual-tree wavelet frames)
analyse (multi-)fractalty
analyse frequencies (with Gabor wavelets for instance)


Due to the redundancy, and the quantity of available wavelets (not the same is best for different purposes), they could  appear a little less efficient for the analysis of pure stationary and harmonics signals, for which Fourier is better suited.

The main drawbacks are:


for fine analysis, it becomes computationaly intensive 
its discretization, the discrete wavelet transform (comp. efficient), is less efficient and natural
it  take some energy to invest in wavelets to become able to choose the proper ones for a specific purpose, and to implement it correcly.