SliverAppBar with TabBar

FlexibleSpaceBar is stacked behind the toolbar and the tabbar. https://docs.flutter.io/flutter/material/SliverAppBar/flexibleSpace.html

You should put your title in SilverAppBar:title not in flexibleSpace.

In your code replace

flexibleSpace: new FlexibleSpaceBar(
                      title: new Text("Some title"),
               ),

with

title: new Text("Some title"),

I had the same problem weeks ago and I could solve using SliverAppBar , FlexibleSpaceBar and SliverPersistentHeader.

Check my code

  @override
    Widget build(BuildContext context) {
      return Scaffold(
        body: DefaultTabController(
          length: 2,
          child: NestedScrollView(
            headerSliverBuilder: (BuildContext context, bool innerBoxIsScrolled) {
              return <Widget>[
                SliverAppBar(
                  expandedHeight: 200.0,
                  floating: false,
                  pinned: true,
                  flexibleSpace: FlexibleSpaceBar(
                      centerTitle: true,
                      title: Text("Collapsing Toolbar",
                          style: TextStyle(
                            color: Colors.white,
                            fontSize: 16.0,
                          )),
                      background: Image.network(
                        "https://images.pexels.com/photos/396547/pexels-photo-396547.jpeg?auto=compress&cs=tinysrgb&h=350",
                        fit: BoxFit.cover,
                      )),
                ),
                SliverPersistentHeader(
                  delegate: _SliverAppBarDelegate(
                    TabBar(
                      labelColor: Colors.black87,
                      unselectedLabelColor: Colors.grey,
                      tabs: [
                        Tab(icon: Icon(Icons.info), text: "Tab 1"),
                        Tab(icon: Icon(Icons.lightbulb_outline), text: "Tab 2"),
                      ],
                    ),
                  ),
                  pinned: true,
                ),
              ];
            },
            body: Center(
              child: Text("Sample text"),
            ),
          ),
        ),
      );
    }

And this is the implementation of SliverPersistentHeader

  class _SliverAppBarDelegate extends SliverPersistentHeaderDelegate {
    _SliverAppBarDelegate(this._tabBar);

    final TabBar _tabBar;

    @override
    double get minExtent => _tabBar.preferredSize.height;
    @override
    double get maxExtent => _tabBar.preferredSize.height;

    @override
    Widget build(
        BuildContext context, double shrinkOffset, bool overlapsContent) {
      return new Container(
        child: _tabBar,
      );
    }

    @override
    bool shouldRebuild(_SliverAppBarDelegate oldDelegate) {
      return false;
    }
  }

I wrote a post about it if you want to check :

https://medium.com/@diegoveloper/flutter-collapsing-toolbar-sliver-app-bar-14b858e87abe

You can use titlePadding for padding:

              SliverAppBar(
                  pinned: true,
                  expandedHeight: 250,
                  primary: true,
                  forceElevated: true,
                  flexibleSpace: FlexibleSpaceBar(
                    titlePadding: EdgeInsets.only(bottom: 62),
                    title: Text(data.name),
                    centerTitle: true,
                    background: data.logo != null
                        ? Padding(
                            padding: const EdgeInsets.only(bottom: 46),
                            child: Container(
                              color: Colors.black,
                            ),
                          )
                        : null,
                  ),
                  bottom: TabBar(
                    tabs: [
                      Tab(text: "Tab1"),
                      Tab(text: "Tab2"),
                      Tab(text: "Tab3"),
                      Tab(text: "Tab4"),
                    ],
                  ),
                ),

class Menu extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return DefaultTabController(
      length: 2,
      child: SafeArea(
        child: Scaffold(
          body: Consumer<MenuBlock>(
              builder: (context, appController, child) {
            return CustomScrollView(
              slivers: <Widget>[
                SliverAppBar(
                  floating: true,
                  pinned: false,
                  snap: true,
                  forceElevated: true,
                  elevation: 1,
                  expandedHeight: 50.0,
                  title: Text('TITLE'),
                  actions: <Widget>[
                    IconButton(
                      icon: const Icon(Icons.add_circle),
                      tooltip: 'Add new entry',
                      onPressed: () {/* ... */},
                    ),
                  ],
                  bottom: TabBar(
                    tabs: [
                      Tab(text: "TAB1"),
                      Tab(text: "TAB2"),
                    ],
                  ),
                ),
                SliverList(
                  delegate: SliverChildListDelegate(
                    <Widget>[
                      Container(
                        // or SizedBox, etc.
                        height: 4200,
                        child: TabBarView(
                          children: [
                     
                        ),
                      ),
                    ],
                  ),
                ),
              ],
            );
          }),
        ),
      ),
    );
  }
}