How to implement a bottom navigation drawer in Flutter

You can use BottomSheet . (Thanks to westdabestdb)

Working on flutter_gallery demo app:

class ModalBottomSheetDemo extends StatelessWidget {
  static const String routeName = '/material/modal-bottom-sheet';

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text('Modal bottom sheet'),
        actions: <Widget>[MaterialDemoDocumentationButton(routeName)],
      ),
      body: Center(
        child: RaisedButton(
          child: const Text('SHOW BOTTOM SHEET'),
          onPressed: () {
            showModalBottomSheet<void>(context: context, builder: (BuildContext context) {
              return Container(
                child: Padding(
                  padding: const EdgeInsets.all(32.0),
                  child: Text('This is the modal bottom sheet. Tap anywhere to dismiss.',
                    textAlign: TextAlign.center,
                    style: TextStyle(
                      color: Theme.of(context).accentColor,
                      fontSize: 24.0
                    )
                  )
                )
              );
            });
          }
        )
      )
    );
  }
}

Building up on westdabestdb's answer and this article: If you want a bottom navigation drawer that is not blocking, with rounded corners and that is not depending on a black background, try this:

class GoogleMapsHomeUI extends StatelessWidget {
final GlobalKey<ScaffoldState> scaffoldKey = GlobalKey<ScaffoldState>();

@override
Widget build(BuildContext context) {

  showBottomDrawer();

  return Scaffold(
    key: scaffoldKey, // use the key to reference the Scaffold from outside
    ... the rest of your background Scaffold
  );
}


void showBottomDrawer() {
// the next line is needed, because the Scaffold needs to be finished before this function continues.
WidgetsBinding.instance.addPostFrameCallback((_) {
  PersistentBottomSheetController? bottomSheetController = scaffoldKey.currentState?.showBottomSheet((BuildContext context) {
  return Container(
        height: 300,
        decoration: BoxDecoration(
          borderRadius: BorderRadius.only( // makes the round corners
            topLeft: Radius.circular(20.0),
            topRight: Radius.circular(20.0),
          ),
          color: Color(0xff232f34),
        ),
        child:  Center(
          child: Column(
            mainAxisSize: MainAxisSize.min,
            crossAxisAlignment: CrossAxisAlignment.end,
            children: <Widget>[
              Container(
                height: 36,
              ),
              SizedBox(
                  height: (56 * 6).toDouble(),
                  child: Container(
                      decoration: BoxDecoration(
                        borderRadius: BorderRadius.only(
                          topLeft: Radius.circular(16.0),
                          topRight: Radius.circular(16.0),
                        ),
                        color: Color(0xff344955),
                      ),
                      child: Stack(
                        alignment: Alignment(0, 0),
                        children: <Widget>[
                          Positioned(
                            top: -36,
                            child: Container(
                              decoration: BoxDecoration(
                                  borderRadius:
                                  BorderRadius.all(Radius.circular(50)),
                                  border: Border.all(
                                      color: Color(0xff232f34), width: 10)),
                              child: Center(
                                child: ClipOval(
                                  child: Image.network(
                                    "https://i.sstatic.net/S11YG.jpg?s=64&g=1",
                                    fit: BoxFit.cover,
                                    height: 36,
                                    width: 36,
                                  ),
                                ),
                              ),
                            ),
                          ),
                          Positioned(
                            child: ListView(
                              physics: NeverScrollableScrollPhysics(),
                              children: <Widget>[
                                ListTile(
                                  title: Text(
                                    "Inbox",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.inbox,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                                ListTile(
                                  title: Text(
                                    "Starred",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.star_border,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                                ListTile(
                                  title: Text(
                                    "Sent",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.send,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                                ListTile(
                                  title: Text(
                                    "Trash",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.delete_outline,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                                ListTile(
                                  title: Text(
                                    "Spam",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.error,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                                ListTile(
                                  title: Text(
                                    "Drafts",
                                    style: TextStyle(color: Colors.white),
                                  ),
                                  leading: Icon(
                                    Icons.mail_outline,
                                    color: Colors.white,
                                  ),
                                  onTap: () {},
                                ),
                              ],
                            ),
                          )
                        ],
                      ))),
              Container(
                height: 56,
                color: Color(0xff4a6572),
              )
            ],
          ),
        ),
      ),
     ); // Container
    },  
    backgroundColor: Colors.black,
    elevation: 5,
    shape: RoundedRectangleBorder(
      borderRadius: BorderRadius.only(topLeft: Radius.circular(15.0), topRight: Radius.circular(15.0)),
    ), clipBehavior: null,
    enableDrag: true,
);

result (I'm not finished with the background yet, but you can interact with the background, while the drawer is open):