Error: Only static members can be accessed in initializers what does this mean?

class AppHomeState extends State<AppHome> with SingleTickerProviderStateMixin {

    TabController _tabController;
    final filterController = new TextEditingController(text: "Search");
    TextFormField email = ...

... is an initializer and there is no way to access this at this point.
Initializers are executed before the constructor, but this is only allowed to be accessed after the call to the super constructor (implicit in your example) was completed.
Therefore only in the constructor body (or later) access to this is allowed.

This is why you get the error message:

controller: filterController,

accesses this.filterController (this is implicit if you don’t write it explicit).

To work around your issue (assuming email needs to be final) you can use a factory constructor and a constructor initializer list:

class AppHomeState extends State<AppHome> with SingleTickerProviderStateMixin {
  factory SingleTickerProviderStateMixin() => 
      new SingleTickerProviderStateMixin._(new TextEditingController(text: "Search"));

  SingleTickerProviderStateMixin._(TextEditingController textEditingController) : 
      this.filterController = textEditingController,   
      this.email = new TextFormField(
        keyboardType: TextInputType.emailAddress,
        controller: textEditingController);

  TabController _tabController;
  final filterController;
  final TextFormField email;

or when the email field does not need to be final email can be initialized in the constructor initializer list:

class AppHomeState extends State<AppHome> with SingleTickerProviderStateMixin {

  SingleTickerProviderStateMixin() {
    email = new TextFormField(
        keyboardType: TextInputType.emailAddress,
        controller: filterController,
    );
  }

  TabController _tabController;
  final filterController = new TextEditingController(text: "Search");
  TextFormField email;

but in Flutter widgets initState is usually used for that

class AppHomeState extends State<AppHome> with SingleTickerProviderStateMixin {

  @override
  void initState() {
    super.initState();
    email = new TextFormField(
        keyboardType: TextInputType.emailAddress,
        controller: filterController,
    );
  }

  TabController _tabController;
  final filterController = new TextEditingController(text: "Search");
  TextFormField email;