Material UI – How can I style the scrollbar with CSS in JS?

Since you want to do this globally, you may want to follow what CssBaseline does in it source code: https://github.com/mui-org/material-ui/blob/master/packages/material-ui/src/CssBaseline/CssBaseline.js

const styles = theme => ({
  '@global': {
    '*::-webkit-scrollbar': {
      width: '0.4em'
    },
    '*::-webkit-scrollbar-track': {
      '-webkit-box-shadow': 'inset 0 0 6px rgba(0,0,0,0.00)'
    },
    '*::-webkit-scrollbar-thumb': {
      backgroundColor: 'rgba(0,0,0,.1)',
      outline: '1px solid slategrey'
    }
  }
});

it looks like the top-level/global selector to use is @global.