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One booking is the whole question
Compensation is measured by how late you arrive at your final destination on a single booking. Everything follows from that.
So a 40-minute departure delay from Manchester that causes you to miss a Frankfurt connection and land in Singapore eleven hours late is an eleven-hour claim on a long-haul band, not a 40-minute nothing. The Court of Justice confirmed this treatment of connecting flights under one reservation, and it is the single most valuable thing to understand about these rules.
The reverse also holds, and it is where people lose. Book Manchester to Frankfurt and Frankfurt to Singapore as two separate tickets, and they are two separate journeys. Your first flight was 40 minutes late, which is not claimable, and missing the second is legally your problem. No compensation, no duty to rebook you, nothing. Self-connecting is cheaper for a reason.
Which airline pays
The operating carrier of the leg that caused the problem. On a through-ticket with two different airlines, that is the one whose delay broke the connection, not the one that was supposed to fly you onwards. Claim against the first.
Distance is measured end to end
Not leg by leg. The band comes from the great-circle distance between your original departure airport and your final destination, which is why a short first leg can still sit in the highest band. Manchester to Singapore is well over 3,500 km however the journey was broken up, so the claim is in the top band.
When the connection is inside the EU but the journey is not
Scope is set by where the journey starts and who operates it, not by where you changed planes. A single booking from Paris to Los Angeles connecting in London is a flight departing the EU, so EU261 covers the whole journey. A single booking from Los Angeles to Paris connecting in London on a US carrier is not covered going out, because it leaves the US on a non-EU airline — a point that surprises people who assume the European leg brings them inside the rules.
What to put in the claim
State your scheduled arrival at the final destination, your actual arrival there, and the flight number of the leg that went wrong. Do not lead with the length of the first delay, because that is the number the airline would prefer to assess. Lead with the arrival gap at the end of the journey.
Work out your own figure
Four regimes checked at once, no sign-up, and it shows you the article the number comes from.