Question

It seems advantages of using Partial NotInheritable Class is that one can declare/implement shared methods in them.

Here is a code sample, to help visualize the matter.

Namespace MyNamespace.Utility
    ' Cannot Declare Shared Function
    Public NotInheritable Class Document
        ' Can Declare Shared Function        
    End Class
End Namespace

This is opposed to

Namespace MyNamespace
    Partial Public NotInheritable Class Utility
        ' Can Declare Shared Function
        Partial Public NotInheritable Class Document
            ' Can Declare Shared Function
        End Class
    End Class
End Namespace

Answer 1

Utility should be a namespace because it is acting as a logical grouping.
Microsoft .NET Framework Nested Types Guidelines

I don’t follow your claim that an advantage of a sealed class is that you can define static methods, any class can do that.