How can one tell if a cyclic molecule is planar???

How can we tell if any one of these molecules is planar?

None of them are planar. In general the only planar organics are hydrocarbons constructed entirely with sp2 carbons and heterocycles which can fit into the sp2 scheme.

In your structures there are many sp3 carbon atoms, each of which cannot be planar.

So I take it that cyclic (or any) molecules can't be planar if there are single bonds present, since only sp2 hybridised carbons (double-bonded) can be planar?

Basically, yes, and including heterocyles that fit into the scheme such as furan, etc.

Thank you.

The only non sp2 containing flat rings are 3 membered ones, which, for obvious reasons, cannot be puckered.