My application has many drag and drop features. While dragging I want the cursor to change to some grab cursor. Internet Explorer and Firefox work fine for this, but Chrome
Try turning off text selection event.
document.onselectstart = function(){ return false; }
This will disable any text selection on the page and it seems that browser starts to show custom cursors.
But remember that text selection will not work at all, so it's the best to do it only while dragging, and turn it on again just after that. Just attach function that doesn't return false:
document.onselectstart = function(){ return true; }