Fast ADC sampling at random intervals with a Zynq?

[KrudyZ]

Except for one shots, scopes are designed to show signals which are repetitive.  What is  displayed is an overlay of a short window in time whose start is either periodic in the case of recurrent sweep "auto" mode or which is periodic relative to some arbitrary trigger.

That is of course the crux here. All modern scopes are "one shot" or real time sampling.
The only exception are equivalent time sampling scopes for the highest possible bandwidth requirements, in which case the signal needs to be periodic, since the actual sample rate is rather low.
So what you are proposing is really a subsampling scope that will be able to restore a repetitive waveform from a smaller set of samples than traditional sampling scopes.
While this is definitely an interesting approach, I don't see a commercial market for it. Existing sampling scopes do an excellent job and don't have any need to reduce ADC or storage speed requirements. All the magic is in the sampler and the time base. The conversion and data processing takes place at a very leisurely pace.
More importantly for any market interest, sampling scopes are a tiny fraction of all scope, precisely because they lack universal usefulness. They simply can't see the glitches that we need to see in digital systems.
The argument that compressive sampling will see some of the glitches some of the time is true, but it would take a much longer measurement time to even find a single one.

[rhb]

That is of course the crux here. All modern scopes are "one shot" or real time sampling.

They are only one shots if you press the "single" button.  In "normal" mode they try to emulate an analog scope which is very much limited to repetitive signals.  Back in the day an analog Tektronix storage scope was the only way to capture an aperiodic signal.  Those were very expensive so very few people had them.

Try an Amazon advanced search with " compressive sensing " as the keywords.  There are roughly 3-4 pages of monographs  after allowing for repetitions that have come out in the last 5 years.  That's a lot of graduate level courses in sparse L1 pursuits.  This is a cutting edge topic in academia.

You are making assumptions about compressive sensing which are not valid.  You are imposing sampling conditions which do not meet the requirements for compressive sensing and then saying it won't work.   If you insist on putting water in the tank of your car, it won't run.  But it's not because there is anything wrong with the car.  You have violated the operating requirements of the engine.

 If you read the literature you will learn how it works and what the correct requirements are.  If you rely upon your  interpretation of a very simplified overview by me you will not.

[KrudyZ]

That is of course the crux here. All modern scopes are "one shot" or real time sampling.

They are only one shots if you press the "single" button.  In "normal" mode they try to emulate an analog scope which is very much limited to repetitive signals.

Not at all. The only difference between single and normal mode on a real time scope is that in single mode it stops after the first trigger.
In normal mode it continues to trigger and on modern scopes it then builds a data base based on the persistence settings, which will show single shot events of ANY of the triggered traces.
They are overlayed for equivalent time correlation, but the odd one out is easily visible, which is what makes this useful in practice.