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With the various clocks I've had and built for CDP's I've definitely found that the power supply makes a difference. I realise that better oscillators with real interesting parameters quoted like phase noise will ultimately out perform something like the eBay oscillators with nonsense performance figures quoted. I would say that the quality and suitability of the supply rail will allow an oscillator to perform optimally.
Any difference a supply would make would almost likely be in the noise floor. We have not found (with our uber-exotic test equipment) all that much difference in supplies. But, you must note that we do not measure anything above 1 kHz, when it comes to jitter frequency. Which is the area one would be looking, if they were interested in the noise floor.