How do I centre an object when placing it on top of another, i.e. placing a statue on a fountain base?
There is, in fact, a "magic centering tool" for this

It's called CTRL-X (advanced move + duplicate).
NOTE: This only works with non-gridded parts, but it works with all non-gridded parts in both the scenery and building menus. It goes like this:
1. Place the fountain base (which is a non-gridded part) where you want it.
2. With the fountain base selected, hit CTRL-X. This creates a copy of the fountain base that's co-located with the original, and also activates the Advanced Move handles to allow you to move the copy on each axis individually. Pull the copy vertically up off the original fountain base a little so you can see the copy clearly.
3. Without hitting the check mark in the center of the Advanced Move UI thingy, move the cursor down to the scenery tab and find the page with the stature y ou want. Click on this statue's icon. When you do this, the copy of the fountain base will change into the statue, and the statue will still have the Advanced Move handles on it.
4. Use the Advanced Move handles to position the statue properly on top of the fountain base.
5. Profit
This is just an example of this very useful technique. If you have a non-gridded part selected and hit CTRIL-X, you clone it and the clone will be in the exact same place as the original. You can then pull the copy out where you can see it and change it into any other non-gridded part you want, then move it where you want. As long as you're careful which Advanced Move handle you pull, the copy (and whatever you subsequently change it into) will be centered on the original.