Just adding on to the advice from shuigao, lokijuh and others, for the LHR-SIN flights, SQ operates A380's with the 2006 generation economy product (2006 Y), as well as longhaul 777-300ERs, which have either the 2006 generation economy product (2006 Y, with aircraft designated 77W) or the newer 2013 generation economy product (2013 Y, with aircraft designated 77WN or 77WR).

In other words, for the LHR-SIN sector:

A380 - 2006 Economy
777 - 2013 Economy on many flights, although occasionally you might get the 2006 Economy seat.

A380 or 777?
Between A380 and 777-300ER with the 2006 generation economy product, both are about the same. It is a very good long haul product. No longer the newest, but still very spacious and comfortable seat, with good IFE. The newest 2013 generation economy seat is currently available on many/most 777's on the LHR-SIN sector, but some will still have the 2006 seat.

Newest 2013 economy or 2006 economy?
The newest 77WN's and 77WR's flying between LHR and SIN have a very new 2013 generation economy seat, which to me is a little more comfortable than the 2006 Y seat on the A380 and 77W. The IFE is also newer and slightly better (Picture quality, selection etc).

If you want the absolute best and newest seat and IFE, then try to get onto the 77WN/77WR flights with this latest product. shuigao has good advice on how to check for this.

Comparing 777 and A380 cabins
SQ's 777's (All types) are equipped 9-abreast 3-3-3 in economy, which is a spacious layout compared to some operators such as EK who operate 777's in 10-abreast 3-4-3. It is a very comfortable and spacious layout, so you have just as much personal space on the 777 Y seat as the A380 Y seat.

That said, the A380 main deck (10-abreast 3-4-3 in economy) is huge and very feels very spacious, especially if your flight is lightly loaded. Some pax also like the 2-4-2 economy A380 upper deck mini cabin, on the flights where it is still offered.

Emex rows
The emex row seats on both A380 and 77W/77WN/77WR have lots of legroom, although you need to pay extra to book them.

More details

For more info on the seats and flights, there are a couple of trip reports with details on SQTalk:

SQ319 LHR-SIN on 77WN with 2013 Y seats

SQ321 LHR-SIN on A380 with 2006 Y seats

Note that since the time those flights were flown, there may have been some changes to catering, amenities and service format, for example simplified menus without itemized beverage lists.

SIN-JKT sector

Note that for your connecting sector to Jakarta, SQ sometimes flies their new A350. If schedule permits, you might want to try out the new economy seat and IFE and pick flights operated by the A350.