Open drawer on clicking AppBar

Alternative to the accepted answer which does not require a GlobalKey:

class _TestState extends State<Test> {
  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      drawer: new Drawer(),
      appBar: new AppBar(
        leading: Builder(
        builder: (context) => IconButton(
            icon: new Icon(Icons.settings),
            onPressed: () => Scaffold.of(context).openDrawer(),
          ),
        ),
      ),
    );
  }
}

Use a Key in your Scaffold and show the drawer by calling myKey.currentState.openDrawer(), here is a working code:

import "package:flutter/material.dart";

class Test extends StatefulWidget {
  @override
  _TestState createState() => new _TestState();
}

class _TestState extends State<Test> {
  final GlobalKey<ScaffoldState> _scaffoldKey = new GlobalKey<ScaffoldState>();

  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      key: _scaffoldKey,
      drawer: new Drawer(),
      appBar: new AppBar(
        leading: new IconButton(
          icon: new Icon(Icons.settings),
          onPressed: () => _scaffoldKey.currentState.openDrawer(),
        ),
      ),
    );
  }
}