How can I show a floating button over all pages in Flutter?

You can extract the scaffold as a layout that contains a bottom sheet, and use this layout in every page you build, passing in the title, body, etc., so that the bottom sheet is persistent in all pages. Snippet below.

import 'package:flutter/material.dart';

void main() {
  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      debugShowCheckedModeBanner: false,
      title: 'Persistent Bottom Sheet',
      theme: ThemeData(
        primarySwatch: Colors.blue,
        brightness: Brightness.dark,
      ),
      initialRoute: "/",
      routes: {
        "/": (context) => Home(),
        "/library": (context) => Library(),
      },
    );
  }
}

class Home extends StatelessWidget {
  Home({Key? key}) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return Layout(
      title: "Home",
      body: Center(
        child: Text("Home"),
      ),
      actions: <Widget>[
        InkWell(
          onTap: () {
            Navigator.of(context).pushNamed("/library");
          },
          child: Tooltip(
            message: "Go To Library",
            child: Padding(
              padding: const EdgeInsets.all(12),
              child: Icon(Icons.library_music),
            ),
          ),
        )
      ],
    );
  }
}

class Library extends StatelessWidget {
  Library({Key? key}) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return Layout(
      title: "Library",
      body: Center(
        child: Text("Library"),
      ),
    );
  }
}

class Layout extends StatelessWidget {
  final String title;
  final Widget body;
  final List<Widget>? actions;

  const Layout({
    Key? key,
    required this.title,
    required this.body,
    this.actions,
  }) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        centerTitle: true,
        title: Text(title),
        actions: actions,
      ),
      body: body,
      bottomSheet: Container(
        width: double.infinity,
        padding: const EdgeInsets.all(15),
        color: Theme.of(context).cardColor,
        child: Text("Persistent Bottom Sheet"),
      ),
    );
  }
}