How to add rounded border for PopupMenu in Flutter?

Another simple way is:

shape: ContinuousRectangleBorder(
         borderRadius: BorderRadius.circular(30),
       ),

You just add like this at PopupMenuButton

shape: RoundedRectangleBorder(
          borderRadius: BorderRadius.all(
               Radius.circular(20.0),
          ),
),

Example

   PopupMenuButton(
      child: Text("Show Popup Menu"),
      shape: RoundedRectangleBorder(
          borderRadius: BorderRadius.all(Radius.circular(15.0))
      ),
      itemBuilder: (context) => [
        PopupMenuItem(
          child: Text("pub.dev"),
        ),
        PopupMenuItem(
          child: Text("Flutter"),
        ),
        PopupMenuItem(
          child: Text("Google.com"),
        ),
        PopupMenuItem(
          child: Text("https://blogdeveloperspot.blogspot.com"),
        ),
      ],
    ),

Adding to the answer given by @Taz You can use themes to have settings for rounded corners in each and every Popups at one place:

MaterialApp(
  // ....
  theme: ThemeData(
  // ....
  popupMenuTheme: PopupMenuThemeData(
    shape: RoundedRectangleBorder(borderRadius: BorderRadius.circular(8))
  )
)