London – Moscow – Beijing – Hanoi – Saigon – Bangkok – Singapore (or vice versa
If you have the time (we’re talking a minimum of 3½ weeks one-way), you can travel from London to Singapore overland, see the route map here. The links below cover travel in either direction, from London or to London:
Step 1: London to Moscow by train. Daily departures, 2 nights, from around £200 one-way with sleeper. Spend at least 1 night in Moscow.

Step 2: Moscow to Beijing by Trans-Siberian Railway Two trains a week, 6 nights, from around £500 one-way in 4-bed sleeper. Spend at least 1 night in Beijing.

Step 3: Beijing to Hanoi by train Two trains a week, 2 nights, about $320 or £220 one-way in soft sleeper. Spend at least 1 night in Hanoi.

Step 4: Hanoi to Saigon by train Several trains daily, 2 nights. Why not stop off to see Hue or Hoi An? About $90 or £50 one-way in soft sleeper.

Step 5: Saigon-Phnom Penh by bus Daily, 6 hours, $12. Spend at least 1 night in PP.

Step 6: Phnom Penh-Battambang by train or bus, bus to the frontier, train to Bangkok.

Step 7: Bangkok to Singapore by train. Daily, 2 nights, from £40 or $80 one-way including sleeper for both nights.

How to arrange this trip…
There aren’t any travel agencies who can arrange the whole trip, so you will need to plan it out and arrange each stage of the journey yourself. It’s an exercise in project management! Unless time is absolutely no object, you should book the key sections in advance through various travel agencies.
Book London-Moscow through a UK European train ticketing agency such as DB’s UK office or europeanrail.com as shown on the London to Russia page;
Book Moscow-Beijing & Beijing-Hanoi through a local Russian agency such as Svezhy Veter or Real Russia as shown on the Trans-Siberian page & Vietnam page.
Tickets for other parts of the trip, for example, Hanoi-Saigon-Phnom Penh-Bangkok can all be bought locally, as you go along. You’ll need to pre-arrange visas for Belarus, Russia, possibly Mongolia, China & Vietnam, and in many ways complying with the various visa requirements (and in some cases, requirements for confirmed onward tickets to be held) is actually the biggest hassle, not buying the tickets for the trains, so check this out carefully using the relevant embassy websites.
Where do you start? First, read through the seat61 pages linked above. Then sketch out your itinerary using a simple spreadsheet like this, deciding where and for how long you want to stop off. Next, check out the visa situation for each country. Finally, follow the advice on each seat61 page to buy tickets for each train journey that you want to pre-book.
Some inspiration…www.rayis.me