You should consider and understand a use for which scopes are uniquely suitable: assessing signal integrity. Understanding why decent manufacturers prominently show eye-diagrams in their advertising will help in that respect.
Isn't that just equivalent-time sampling rate? I don't mean per se, but rather the principle behind it. Since you have a repetitive waveform you just sample a bunch of times (eye diagrams do it randomly, don't they?) and you don't need a high sampling rate.
To all intents and purposes, yes. There are a number of variants, but they don't affect the point.
The key point is to disassociate MHz and S/s in people's minds; they are orthogonal concepts. People that conflate them often get the wrong answer to "if I have a 10kHz signal on a 10MHz carrier, how fast must I sample in order to see the signal?". As I mentioned earlier, I have ~2.5GHz scope with a ~10kS/s rate.
Your scope is a "sampling" oscilloscope and can not be used for one-shot signals. It absolutely requires a stable, repetitive signal in order to generate a waveform over a relatively long period of time, spanning hundreds to many thousands of cycles of that waveform.
Correct.
This is a special purpose instrument and can be very good at what it can do, but there is a long list of things that it can't do as a result of its principle of operation.
That is true of all scopes, and indeed of all instruments.
Its limitations are similar to those of an analog CRT oscilloscope.
Nonsense. Clearly your experience is limited. ...
You don't need to resort to attacks.
It is not nonsense. Or show me how it is. I chose my wording carefully. "similar to", not "identical to".
My experience is not "clearly limited".
... Start by considering analogue scopes that store single-shot waveforms without digitising them.
Analog storage CRT oscilloscopes are exceptional and very uncommon. The vast majority of analog oscilloscopes ever produced required a repetitive waveform to display a static image. Either that or a Polaroid camera attachment to capture a one-shot. Ask anyone who did
that back in the day if they'd rather have a modern real time scope with advanced triggering capabilities instead.
Due to improvements in ADC and memory speed, most oscilloscopes these days are "real time" oscilloscopes. A real time scope can capture and display a waveform in one shot (one trigger), at its full bandwidth. This is the only way to capture and view non-repetitive signals. Only the fastest (10's of GHz) scopes will use equivalent time sampling, though many higher end scopes are real-time with an optional equivalent time sampling mode for better detail on high speed repetitive signals.
True; the advances in ADCs over the past 15 years has been remarkable. But that doesn't affect the point.
For example, if you want to analyze a clock on your digital board, you can use a sampling scope. If you want to use your scope to view or analyze a serial data stream, you'll need a real time scope.
I wonder how we managed for the decades before decent ADCs became available? Oh, yes, we used skill and imagination; cue the aphorism in my .sig 
In particular, we used scopes and eye diagrams to verify (analogue) signal integrity. Once that was assured, we flipped to debugging in the digital domain with logic analysers and printf statements. That's still a very sound development strategy partly because it works at all speeds not merely the trifling slow-speed stuff like MCUs.
Note again my carefully chosen wording: "If you
want to use your scope to view or analyze a serial data stream, you'll need a real time scope."
I did not suggest this is the only or best tool for the job, but if it is the tool that you have at hand, then it can be the right tool.
Another example, you might have two uncorrelated signals, like a modulated signal, or an amplifier output that is oscillating. You may not see that with the sampling scope as the oscillations won't be in phase with the signal the scope is triggering on. At best it will show noise on top of a signal.
If you are looking at modulation, you should consider modulation domain analysers
If you are looking at oscillators, scopes are likely to be unrevealing: you need a spectrum analyser with very good phase noise characteristics.
Choose the appropriate tool for the job at hand.
Appropriate tools these days
include scopes that sample fast enough to be useful in much broader usage scenarios than the limited scopes of yesteryear.
The budget for a "spectrum analyzer with very good phase noise characteristics" is well beyond most hobbyists, especially those participating in a Beginners forum. A modulation domain analyzer is another special tool which does one thing very very well. The rest of the time it is just an expensive shelf queen. A fancy high speed logic analyzer would be great to have, but again it is expensive and takes up a pile of space. The average Beginner who is considering the likes of a low end Rigol scope probably doesn't need or want such things. One carefully chosen tool in the hands of someone who understands it capabilities
and limitations can cover a broad set of measurement tasks. Maybe you lost my point about the limitations of sampling scopes compared to real time ones. Or maybe you just love your scope so much that it biases your opinion. I get that, really I do. But let's be pragmatic here. If a person has only one scope on the bench, they would be poorly served by a low speed, high bandwidth sampling scope, especially with all the affordable options in 2017.
Oh, and I didn't say anything about oscillators.
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