vue-loader@15.*, you need to configure css-loader separately.

const VueLoaderPlugin = require('vue-loader/lib/plugin')

module.exports = {
  // ...
  module: {
    rules: [
      {
        test: /\.vue$/,
        loader: 'vue-loader'
      },
      {
        test: /\.css$/,
        use: ['style-loader','css-loader']
      }
    ]
  }
  plugins: [
    new VueLoaderPlugin()
  ]
}

Vue Loader v15 now requires an accompanying webpack plugin to function properly:

// webpack.config.js
const VueLoaderPlugin = require('vue-loader/lib/plugin')

module.exports = {
  // ...
  plugins: [
    new VueLoaderPlugin()
  ]
}