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As with a lot of things, there isn’t a “best” thing as this is a very subjective matter but we can focus on the “most popular”. I will give you an overview of the current state of most popular Vue.js styling solutions:
Vuetify
A complete material design UI framework has a lot of components and is very active, very customizable as well It probably is the most popular UI solution for Vue.js
Quasar Framework
Quasar is also a material based UI framework but it focuses on performance and developer experience, very active and releases often. It also has its own CLI and allows you to target apps using Electron or Cardova, its more of a complete app-bootstrapper than a UI framework.
Bootstrap-Vue
It is like the name suggests, a bootstrap-based UI framework and is the most comprehensive one with full implementation of bootstrap and a lot of custom components as well. It the way to go if you are going with bootstrap-based UIs.
Tailwind CSS
Tailwind is a utility-first CSS framework that uses PostCSS, it is really easy to get started with and very productive, it is my personal favorite because it stays in the CSS and HTML realm, while UI frameworks are very cool and productive, you cannot escape the overhead they introduce and not all designs are based off bootstrap or material anyways. Tailwind manages to stay productive while being very flexible but requires having decent knowledge of CSS and how properties interact.
Again there isn’t a “best” framework out there for styling, so you need to study the options and pick the one that “better suits you”.