I'm not sure what plane-spotting means technically, but if
you're referring to gawking at planes other than from the terminal
buildings, then yes and no.
Yes, it's possible to catch the planes during takeoff and landing
from the perimeter fence. Usually we drive up to one of the
emergency gates and hang around there. You can get really awesome
views of the gigantic monsters of planes. Plus, the noise is really
deafening, but it all adds to the experience.
The downside is that it is illegal to crowd around the emergency
gates for obvious reasons. I've been chased away by police at least
once, but most of the time, nobody bothers you.
Just
don't visit the gate next to the airport fire station.
Just grab a cab and get the driver to bring you to Changi Coast
Road. Make sure he stays there throughout your adventure, or you'd
have problems getting back to the terminal.
SQ22