Can you get better prices by flying without checked-in luggage on intercontinental airlines?

Almost all airlines include some checked baggage for free on all longhaul journeys. Certainly both Emirates and Qatar include a baggage allowance. There may be some exceptions with for instance Norwegian or Air Asia (I've never used them).

So you cannot “deselect” baggage.

The problem with low cost carriers (LCC) is that they almost exclusively offer point-to-point itineraries, and don't offer connecting flights. There are some exceptions though (I know personally that flydubai offers transfers through their hub in Dubai).

The second issue you'll have is that most LCC fly aircraft that do not travel the distances (without refueling) required to get from Europe to Asia (there are some exceptions though, AirAsia for example).

So for the best bang for your buck you'll have to stitch together an itinerary.

First task would be to find out which LCC fly to Asia; or can fly you to another airport from where you can catch another LCC to Singapore.

The first one that came to my mind was Transavia which fly from Amsterdam to Dubai.

As an example, I was able to book:

• Barcelona - Amsterdam (Transavia)
• Amsterdam - Dubai (Transavia)
• Dubai - Lahore (flydubai)
• Lahore - Singapore (Malindo Air)

Hat tip to Zach for pointing out the many risks to this itinerary:

1. You may have to exit the secure zone and check in again (to get the next boarding pass), which means you would need visas for the transit countries.

2. If you miss your connection, then you cannot claim compensation as you are on a point-to-point itinerary.

3. Zach also mentioned that if you end up checking in luggage, it would be your responsibility to collect it (which would require exiting the secure zone) and then re-check it in; however as you are specifically traveling with carry-on luggage, this may not be a concern.

Most connections from Europe to Asia are on airlines with a "traditional" business model where some luggage is included even in the cost of the most basic fare. However I am aware of at least two low-cost-carriers that operate Europe to Asia with fares that have an even lower booking class without luggage (nor included meals):

• Eurowings e.g. from Cologne-Bonn (CGN) to Thailand (Bangkok or Phuket). You would save EUR 50 in this case by booking the "basic" instead of the "smart" fare class.

• Norwegian e.g. from Copenhagen (CPH) or Oslo to Bangkok. Again I picked a flight on a random day and choosing the "lowfare" instead of the "lowfare+" class gave me a savings of USD 90.

To make use of these savings you will need to get to these hubs, which is likely cheapest on an unconnected flight, or with other means (buses, trains, hitching, ...).

An even better possibility might be to do shorter hops on different LCC. On the strip from Europe to Asia this will likely take you via the Arab world, and potentially India/Sri Lanka. Here is a trip report of a Telegraph reporter who did a round-the-world this way.
From Europe to Singapore he did Prague->Dubai->Colombo->Singapore for a total cost of GBP 300 on three different LCC, including flyDubai and AirAsia.
If you follow this path you will also want to consider to stay where you stop, since you are backpacking anyway.