I upgraded babel 6.x → 7.x but having issues running Webpack. 
It is complaining about missing core-js/modules/*. 
My babel.config.js is in the root directory. I converted the previously existing .babelrc to js (.babelrc also produced the same errors). I am guessing it is some collision with all the core, corejs2, runtime stuff. 
There are two apps in my src, mine and Styleguidist (in ./node_modules). My app transpiles and works with these same package.json/babel.config, but Styleguidist does not.
The error when running Styleguidist with webpack:
Module not found: Error: Can't resolve 'core-js/modules/es.array.concat' in '/project/src/node_modules/react-styleguidist/lib/client/rsg-components/Slot'
/node_modules/react-styleguidist/lib/client/rsg-components/Slot.js:
import "core-js/modules/es.array.concat";
import "core-js/modules/es.array.filter";
...
package.json
"dependencies": {
    "@babel/polyfill": "^7.0.0",
    "@babel/runtime-corejs2": "^7.4.3",
}
"devDependencies": {
    "@babel/core": "^7.4.3",
    "@babel/plugin-proposal-class-properties": "^7.0.0",
    "@babel/plugin-proposal-decorators": "^7.0.0",
    "@babel/plugin-proposal-export-namespace-from": "^7.0.0",
    "@babel/plugin-proposal-function-sent": "^7.0.0",
    "@babel/plugin-proposal-json-strings": "^7.0.0",
    "@babel/plugin-proposal-numeric-separator": "^7.0.0",
    "@babel/plugin-proposal-object-rest-spread": "^7.4.3",
    "@babel/plugin-proposal-throw-expressions": "^7.0.0",
    "@babel/plugin-syntax-dynamic-import": "^7.0.0",
    "@babel/plugin-syntax-import-meta": "^7.0.0",
    "@babel/plugin-syntax-jsx": "^7.0.0",
    "@babel/plugin-transform-modules-commonjs": "^7.4.3",
    "@babel/plugin-transform-react-jsx": "^7.3.0",
    "@babel/plugin-transform-runtime": "^7.4.3",
    "@babel/preset-env": "^7.4.3",
    "@babel/register": "^7.0.0",
    "babel-core": "^7.0.0-bridge.0",
    "babel-eslint": "^10.0.1",
    "babel-helper-vue-jsx-merge-props": "^2.0.3",
    "babel-jest": "^24.7.1",
    "babel-loader": "^8.0.0",
    "babel-plugin-dynamic-import-node": "^2.2.0",
    "babel-plugin-transform-vue-jsx": "^4.0.1",
}
babel.config.js
module.exports = {
    presets: ['@babel/preset-env'],
    plugins: [
        '@babel/plugin-transform-runtime',
        '@babel/plugin-transform-react-jsx',
        'transform-vue-jsx',
        "@babel/plugin-proposal-object-rest-spread",
        "@babel/plugin-syntax-dynamic-import",
        "@babel/plugin-syntax-import-meta",
        "@babel/plugin-proposal-class-properties",
        "@babel/plugin-proposal-json-strings",
        [
            "@babel/plugin-proposal-decorators",
            {
                "legacy": true
            }
        ],
        "@babel/plugin-proposal-function-sent",
        "@babel/plugin-proposal-export-namespace-from",
        "@babel/plugin-proposal-numeric-separator",
        "@babel/plugin-proposal-throw-expressions"],
    comments: false
}
 
     
     
     
     
     
    