egui-101-menu
This is a simple example with a menubar and multiple submenus.
Source
Screenshots
 

Code
Typically, a menubar will go on the top panel. We create a menu::bar and populate it with a ui.menu_button. Each of these can contain multiple ui.button.
Here is an example of how to render a menu bar with a single dropdown menu:
egui::menu::bar(ui, |ui| {
    ui.menu_button("File", |ui| {
        if ui.button("Save").clicked() {
            //functionality
        }
        if ui.button("Quit").clicked() {
            std::process::exit(0);
        }
    });
});