TLDR:

Official policy: https://www.singaporeair.com/en_UK/b...question-60958
Note that the requirement of having a single itinerary to transit through SIN is out of date

SQ - SQ
Yes, provided your connection is legal.

SQ - Other airlines
Yes, in most cases. As long as there's an interline agreement with the subsequent carrier(s), and your connection time is legal, SQ should check it through. There may be exceptions, such as where the connecting point cannot accept through-checked bags.

Scoot (TR) - SQ
No, Scoot will not tag your bag to an SQ flight if it's on a separate ticket.

Other airlines - SQ
Check with the airline that will be accepting your bag.

FAQ
How do I ask to be through-checked from SQ to another airline?
At the check-in desk, tell the agent you have an onward flight, and show a printout/screenshot of your e-ticket for the onward flight(s).

Will I receive my boarding pass(es) for the subsequent flight(s)?
Most likely. SQ has the capability of printing boarding passes for your subsequent flights, even on other airlines, but this doesn't always work.

Is SQ obligated to check me in across separate tickets?
No, especially if it's to another airline. It is, however, a courtesy offered by SQ at their discretion and is not guaranteed. There may be instances of inexperienced/unhelpful check-in staff, or unforeseen exceptions. You should always have a contingency plan in case you're refused. In other words, always meet the entry requirements for the destination on your first ticket, and leave sufficient time to clear arrival formalities and re-check your bag.

My bag was tagged by SQ to another airline, and now it's mishandled (delayed/lost/damaged). Which airline is responsible?
The final delivering carrier is always responsible, even if they're not at fault. This is true even if you're on separate tickets. You must file a report with the final carrier listed on your baggage claim stub.

I was through-checked by SQ, but missed my connection. Will SQ rebook my onward travel?
If it's SQ-SQ, they're usually proactive at rebooking you even on separate tickets. If it's to another carrier, you're at the mercy of that airline and your fare rules. This is the inherent risk of travelling on separate tickets, so plan accordingly.

My tickets have different baggage allowances and/or I hold status with the subsequent airline. Which allowance will SQ honour?
Anecdotal reports suggest SQ will usually honour the higher allowance, if your onward flight is on SQ. If your onward flight is on a different airline, expect SQ to honour the allowance of your flight with them, along with any extra allowance your applicable status would get you on SQ (i.e. PPS/KFEG/*A/VS/VA Gold etc). Should you hold status that gets you extra allowance on the subsequent airline, but not SQ (e.g. oneworld Emerald, AC E35K, etc), don't expect SQ to honour it.