Well, I only use up to 15V really, so there are no problems in that area. The transformer itself is small 24VA 230->18VAC, so I am aware that it can't be called "oversized" for more than those 15VDC. Anyway, this is really my first attempt to build a simply power supply and to be frank I am learning basic electronics building it. Anyway, thanks for the info regarding short bursts of energy from transformer. I didn't think about that.
As for my original problem I managed to solve it. tecman (Paul) showed me correct direction where to look and by accident I got another clue when using my scope. At one stage of my 'investigation' I used both channels of my scope. One connected to the DC-DC converter (ground there) and one to the LM317 (ground there). I don't know what is the internal construction of my scope, but I suspect that both probes share the same ground. The problem disappeared when both 'grounds' were shorted.
I believe that I made a classic mistake by not properly wiring ground. I disconnected the scope and measured the potential between those two 'grounds' - it was 0.15V. Shorting them solved the problem and now even Low ESR cap works very good. I don't see any spikes from DC-DC converter at the output of DC regulator. It seems that everything now works as expected.
Question is: did I fix that problem correctly and why the ground drifted so much? I know that these questions are probably trivial for most of you, but for newbie like me it is not so obvious. Again, thanks for your help guys. I really appreciate that.