How to implement a bottom navigation drawer in Flutter

I quickly made it, but you are going to have to implement active page text/icon colors to the listview. Also, the full code is here if you want to copy from the gist.

class ScreenOne extends StatefulWidget {
  @override
  _ScreenOneState createState() => _ScreenOneState();
}

class _ScreenOneState extends State<ScreenOne> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text("Reply demo"),
      ),
      bottomNavigationBar: BottomAppBar(
        elevation: 0,
        color: Color(0xff344955),
        child: Container(
          padding: EdgeInsets.symmetric(horizontal: 10.0),
          height: 56.0,
          child: Row(children: <Widget>[
            IconButton(
              onPressed: showMenu,
              icon: Icon(Icons.menu),
              color: Colors.white,
            ),
            Spacer(),
            IconButton(
              onPressed: () {},
              icon: Icon(Icons.add),
              color: Colors.white,
            )
          ]),
        ),
      ),
    );
  }

  showMenu() {
    showModalBottomSheet(
        context: context,
        builder: (BuildContext context) {
          return Container(
            decoration: BoxDecoration(
              borderRadius: BorderRadius.only(
                topLeft: Radius.circular(16.0),
                topRight: Radius.circular(16.0),
              ),
              color: Color(0xff232f34),
            ),
            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),
                          overflow: Overflow.visible,
                          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.stack.imgur.com/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),
                )
              ],
            ),
          );
        });
  }
}