Low-noise amplification of microphone signals, tubes or chips?
In outline, it depends on the signal source, i.e. the type of microphone.

There are some very low noise vacuum tubes.

There are some low noise IC amplifiers, but not many.

There are also discrete semiconductors, both bi-polar and JFET, and these are often the best choice for an input stage, possibly using an IC for the later gain and output stages.



Among vacuum tubes, the 7586 (Nuvistor) has a good reputation. (Ditto some makes of 6060 triode if I recall correctly, and, I think, the V301 if you have a reliable source of WW2-era components...).

To use a valve as a low noise amplifier, you have to remember that it actually has quite a high noise resistance, so it achieves its best noise figure at a high source impedance. You can after all keep the grid circuit impedances arbitrarily high.

So a valve operated as a cathode follower is a good choice to provide current gain when fed from a capacitor mic capsule (typically 30pf). It is necessary to bias the input with a grid leak resistor in the region of 1 Gohm or higher (consider the RC time constant and you will see this determines the LF performance of the mic). See Neumann U47 etc.

But for a low source impedance such as a ribbon microphone, the only way the valve can achieve low noise performance is by matching that source impedance to the mic's noise impedance with a high ratio step-up transformer.



A good replacement for a vacuum tube in similar applications (high source Z) is a suitable low-noise JFET. Get hold of the NatSemi "Discrete Databook" from 1978 : now a rarity but containing more good info than you can find almost anywhere else - including noise voltages vs frequency and current for a range of devices.

You'll find that some relatively large area FETs (aimed at switching) have relatively low noise voltages. Effectively these are multiple small JFets in parallel. See rms summation of noise sources... Pay special attention to "Process 55" (2N5459) operated at drain currents of 1mA or more.



For low source impedance, as far as I know, you still can't beat bipolar transistors - usually PNP, and usually medium power (again, relatively large area). Even the BC214 isn't bad (see the Nat Semi databook again) but some designers recommend the Hitachi 2SC2547 (and if you need an NPN equivalent, I think the 2SA1075). With these, at currents around 10mA, you can reduce the noise resistance of the transistor to somewhere in the 10-30 ohm range. 

I'll leave you to convert that to nV/rtHz and compare with the best opamps you can find, or the AD797...



Between vacuum tubes and discrete semiconductors, each used to its best, I doubt you'll find more than a dB or so difference in noise level. 

Ultimately, of course, you are bound by the noise figure of the mic capsule itself, i.e. the Brownian motion of the air molecules hitting it. Again, a large area capsule is quieter (sensing a theme here?) at the expense of inferior HF performance (when its dimensions exceed 1/4 wavelength of the sound signal). There have been twin-diaphragm mics (large LF, small HF) with internal crossovers to overcome this downside.

And as @pjc50 notes, any further improvement beyond this comes from, effectively, multiple microphones in parallel : with DSP to not only overcome the disadvantages of their spatial dispersion but also offer advantages such as synthetic beamforming.