In this paper, we used wavelets to detect raindrops based on its shape:

https://www.researchgate.net/publication/299432803_Real-time_raindrop_detection_based_on_cellular_neural_networks_for_ADAS?ev=prf_pub

Article Real-time raindrop detection based on cellular neural networ...

common wavelets act only in 3 directions: horizontal ,vertical and diagonal. In my idea, you could try wavelets who operate on more directions for example " dual-tree wavelets"

Depends on which imaging modality you are working with and what your images look like. In fact, depending on your images, wavelets for texture discrimination might not even be the right approach. You might want to look at gray-level cooccurrence matrices: 

http://haralick.org/journals/TexturalFeatures.pdf

Also, you could also look at histogram-based methods for discrimination. Here is a publication that might interest you:

https://www.ncbi.nlm.nih.gov/pubmed/27819858

As for wavelets and their choices

I advise you to use several types of wavelet mothers with adaptive thresholding

https://www.sciencedirect.com/science/article/abs/pii/S0167865502003902