In .prettierrc, add:

{
  "singleQuote": true,
  "jsxSingleQuote": true
}

I don't have prettier extension installed, but after reading the possible duplicate answer I've added from scratch in my User Setting (UserSetting.json, Ctrl+, shortcut):

"prettier.singleQuote": true

A part a green warning (Unknown configuration setting) the single quotes are no more replaced.

I suspect that the prettier extension is not visible but is embedded inside the Vetur extension.

Use a Prettier configuration file in your project folder: .prettierrc

Inside your config file use Prettier overrides: https://prettier.io/docs/en/configuration.html#configuration-overrides

So in your case this example config should work (.prettierrc):

{
    "singleQuote": true,
    "overrides": [
        {
            "files": ["**/*.css", "**/*.scss", "**/*.html"],
            "options": {
                "singleQuote": false
            }
        }
    ]
}

In addition to @Barmar answer, you could also use prettier configuration on a .eslintrc.js file using the following, property:

rules: {
    // ...
    'prettier/prettier': ['error', { singleQuote: true }]
  }