Wavelet methods serve as a powerful tool in applied mathematics and gaining significant attention in physics, mathematics, and engineering research for their ability to analyze various dynamic phenomena through waves. These methods have proven to be an efficient and powerful in dealing with any kind of differential equations and offer versatile solutions. In this paper, our main aim is to review the currently published work on three kinds (Chebyshev/Legendre/Lucas) of orthonormal/orthogonal wavelets along with the applications. This review of the literature highlights the effectiveness of wavelet methods. They offer advantages such as adaptive resolution, which allows for the localization of features in both time and frequency domains. A quick recall of a straightforward overview of the fundamentals of wavelet transform opens the direction and future scope in proposing wavelet-based scheme.