What challenges are there for more, smaller liquid fuel engines instead of one large?

Question

Some questions popped up about difficulties of creating throttleable liquid fuel engines.

I wonder why the obvious alternative isn't more common: multiple engines, from which only some will be lit; possibly ones of varied thrust so that you can fine-tune your thrust by selecting which engines are active simultaneously.

Several Soviet launchers used four engines. The Shuttles used three engines. Falcon Heavy uses 27 Merlin engines. Soyuz uses 20 combustion chambers across five engines - so the multi-engine solution is not unheard of - but not quite as common as single-engine ones. What factors decide upon its relatively low popularity?

Answer 1

Having a large range of thrust settings is important for landings, not so much for launches. During a launch, you want some throttlability to reduce loads around max-Q, but otherwise a launch is most efficient when it's done as quickly as possible. So in non-reusable launchers, there's no point in being able to shut down some of your engines.
When you use more engines, your parts count, cost and weight goes up and your reliability goes down.
If you have enough engines (about 5 engines seems to be enough), you gain the option to continue the mission if one engine fails, but for most designs that decision hasn't been made (more launchers with a small number of engines than with engine-out capability).

Answer 2

As discussed in this other thread, the biggest problem with having lots of little engines is that lots of little engines means lots of small fuel and oxidiser lines (whereas, if you use a few big engines, you can get away with using a few big fuel and oxidiser lines). Which, in turn, creates two problems:

1. More complicated plumbing makes it a lot harder to ensure that the fuel and oxidiser flow is stable and uniform - especially when you have a lot more engines that all need to be producing the same amount of thrust. As a result, designs with huge numbers of engines often have severe problems with fuel- and oxidiser-flow instability, which leads to combustion instability, which causes the chamber pressure of the engine suffering from it to fluctuate, which affects the amount of fuel and oxidiser that enters the engine, which worsens the instability, all of which induces vibrations (called "pogo oscillation") in the vehicle structure, leading us to the second big issue:
2. Having lots of small propellant lines makes the rocket's plumbing much more fragile - smaller, thinner lines break more easily than bigger ones, and there're more lines that can break. When you have a vehicle that's already prone to severe pogo oscillation, fragile plumbing is not your best friend.

Look at the Soviet N1 rocket for an example of the problems that can arise when using lots of little engines. Of the four N1 launch failures (which comprised the N1's entire operational history), three occurred after parts of the rat's nest of plumbing feeding the 30(!) engines in the N1's first stage ruptured for one reason or another (pogo oscillation in the first launch, an exploding turbopump in the second, the fuel/oxidiser equivalent of a water hammer in the fourth), setting parts of the stage on fire, which rapidly burnt through various important components.

TL;DR: Having lots of small engines means having lots of complicated, fragile plumbing, which is generally ungood for a rocket.