Spherical fuzzy sets (SFSs) have recently become more popular in various fields. It was proposed as a generalization of picture fuzzy sets and Pythagorean fuzzy sets in order to deal with uncertainty and fuzziness information. The similarity measure is one of the beneficial tools to determine the degree of similarity between objects. It has many crucial application areas such as decision making, data mining, medical diagnosis, and pattern recognition. In the short time since their first appearance, some different distance and similarity measures of SFSs have been proposed, but they are limited through the literature. In this study, some novel distances and similarity measures of spherical fuzzy sets are presented. Then, we propose the Minkowski, MinkowskiHausdorff, Weighted Minkowski and Weighted Minkowski-Hausdorff distances for SFSs. In addition, trigonometric and f-similarity measures are developed based on the proposed distances in this paper. The newly defined similarity measures are applied to medical diagnosis problem for COVID-19 virus and results are discussed. A comparative study of new similarity measures was established and some advantages of the proposed work are discussed.