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