spotting can be done from T1 viewing mall. It's through tainted glass, but it will do (a bit of Photoshop).
Otherwise go to Changi beach, oposite of the cargo terminals Superhub 1 or Superhub 2. You can do this by taking the MRT to Tampines MRT station (from the city or from the airport does not matter), take bus 19 to Changi Beach.. Go to either carpark 3 or six carpark 6. It's really taking pictures with your feet in the surf and your camera into the air... From those car parks you get the finals for rwy 20R and 20C. Or take your chance along the fence through the crash gates
For 02L, go to Changi Business Park Central Rad.... you get there via the Tanah Merah MRT station, take bus 47 or 118 to Changi Business Park Bus Terminal. Good for finals 02L