The galaxies are not randomly distributed. In fact there are correlations between galaxy distributions at various scales that essentialy can be traced to initial perturbations that arose after the big bang. Multiple projects are now working on exactly figuring out those correlations as a function of scale.

I understand the galaxies can be traced back to those initial perturbations, but regardless of that, it'd be interesting to see if there are any patterns that can be identified at a large scale and based on similar sky survey images posted here. Are there any projects working on identifying such patterns?

Yes, there is a pronounced filamentary structure to galaxy clusters.

https://en.wikipedia.org/wiki/Galaxy_filament

with a concrete rendering at

https://en.wikipedia.org/wiki/Void_(astronomy)#/media/File:S...

and some more at

https://en.wikipedia.org/wiki/Observable_universe#Walls,_fil...

These filamentary luminous matter distributions (and the sparse, dark, cold voids between them) drive our theories of structure formation:

https://map.gsfc.nasa.gov/universe/bb_cosmo_struct.html

There are some patterns at scales of up to a few tens of megaparsecs, but beyond that matter is for all practical purposes homogeneous and isotropic: with a good enough (infrared- or radio-) telescope you will observe a galaxy or a precursor along every line of sight unobscured by objects within our own galaxy, and only the reddening and dimming from the metric expansion of space permits any darkness at all in our sky.

This is important to the standard mode of cosmology, \Lambda-CDM, which relies on Friedmann equations, which model all the matter content of the entire universe (past and present) as a perfect fluid of uniform dust. Because of the "local" nonuniformities like filaments and voids, grains of dust are thus on the order kilo- to megaparsecs. If we find that there are bigger grains, the standard "concordance cosmology" will have to be adjusted to be concordant with the new large-scale evidence.

(There is lots of looking right now, but so far everything points to smoothness at scales much smaller than gigaparsecs, kinda how a sheet of paper looks smooth in your hands, but a tiny piece of it under a light microscope looks very fibrous, and under an electron microscope one might discern the remnants of plant cells, or even see microbes!).

WMAP went from the cosmic microwave background downwards. So looking only at the largest structures.

The patterns it sees are real, and very cool.

https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy...

I don't know if I understand what you're asking... what meaning would there be in measuring "an odd number of galaxies within a specific radius of a red giant star"? It would be purely from our point of view, those galaxies might look like in a radius in a flat representation but they are incredibly far away from each other and the red giant star it's waaay closer to us than those galaxies. Same kind of reasoning applies to the other examples... could you elaborate a bit?

Not to say you're wrong about anything, I simply didn't get it

edit: grammar

For example we see gravitation lensing effect even though the galaxy that is causing it is in front of what is being lensed from our perspective. So there may still be reasons to see if there are any patterns even if it's just from our perspective and also because it's the only one we have access to.

As sega_sai said, there are patterns in the way the galaxies are distributed. Around every galaxy, there is a ring where there is a greater chance of finding another galaxy. It's called the Baryon Acoustic Oscillation, and measuring it is the reason we took these images in the first place! Measuring the size of that ring at different times in the past will hopefully help us figure out how dark energy behaves. See desi.lbl.gov, eg.