I'm from Singapore and planning a tour of Europe, landing at Paris and taking a train around several Schengen countries before departing at Rome. If I buy a one-way Singapore-Paris ticket and another one-way Rome-Singapore ticket, is that sufficient proof of onward travel out of the Schengen area or do I need a ticket out of France when arriving in Paris?
Schengen countries rely on each other to check travellers who cross an “external border”. What this control entails is defined in the relevant EU regulations.
If you are only there to visit, you should therefore show that you intend to leave the whole Schengen area, and not merely the country you happen to visit first. Leaving France to Italy would not be enough but flying to Singapore from Italy would.
Note that having an actual ticket is not formally required but you might be asked how you plan to leave the area or go to Italy or more generally about your itinerary. If it does come to that, the more you can provide, the better.
According to the Schengen Border Code Article 5
For stays not exceeding three months per six-month period, the entry conditions for third-country nationals shall be the following: [...]
(c) they justify the purpose and conditions of the intended stay, and they have sufficient means of subsistence, both for the duration of the intended stay and for the return to their country of origin or transit to a third country into which they are certain to be admitted, or are in a position to acquire such means lawfully;
So you don't even need to have any ticket booked at all as long as you can convince them that you have the funds to acquire a return ticket to leave the Schengen area.
In essence you should be clear.