How to select current word in Visual Studio Code (VS Code)?

You are looking for Shrink/Expand Selection.

Trigger it with Shift+ Alt+Left and Shift + Alt+Right

Update:

This is now called Smart select API. This feature uses semantic knowledge to intelligently expand selections for expressions, types, statements, classes, and imports.

On Mac OS: Cmd+D
On Windows & Linux: Ctrl+D

Above solved the purpose for me.

But ⌘D is defined as "editor.action.addSelectionToNextFindMatch", so if you press it more than once, it will try to search and select same word in the file which then can be used to do "multi word editing".