I'm struggling to understand what the transit country was, the only obvious European non-Schengen country with a major airline hub I could think of was the UK, but Singaporean nationals don't need a visa to enter the UK.
Anyway, while the posters above me are correct that it is ultimately the passenger's fault for not possessing the correct visa to complete the itinerary, the airline check-in agent at VCE should have checked for a valid Schengen visa as that leg of the multi-city trip terminated in the Schengen area. In this way the OP would at least have not ended up stranded in the transit country but could have booked travel directly home from VCE (or made alternate arrangements to transit at a Schengen area hub).
Agents often don't do this, only checking the required documents for the first leg and this results in the passenger being unduly transported to (and potentially stranded at) an intermediate point.
Since the traveller did not have a visa to enter the transit country, they should only have been transported there if they had valid documents to continue their journey, which they did not.
What the airline effectively did was transport the pax to the transit country without a valid visa to terminate their trip there. Denied boarding should have occurred at VCE, not the transit airport.
For example, if I flying Vancouver-Shanghai-New Delhi using the China TWOV program, the check-in agent in Vancouver should ensure that I have the correct documents to enter India otherwise I am not eligible to transit China without a visa and they are transporting me to China without proper documents.