No pude encontrar esa cosa por ningún lado.

English Translation

I wasn’t able to find that thing anywhere.

Google Ngrams says that “por ningún lado” is more common than “en ningún lado”, but “en ninguna parte” is more common than “por ninguna parte”. Out of the four, “en ninguna parte” is reported as by far the most commonly used (in writing).

I have nothing concrete to sustain this on, but I think that “en” gives the feeling that you have been looking in several individual spots, and couldn’t find whatever you were searching, while “por” is rather about “zones”.
So, perhaps:

  • I have serached under each cushion, in the cupboard, on the table : no lo he encontrado en ningún lado
  • I have looked around in the kitchen, in the livingroom, in each corner of the house: no lo he encontrado por ningún lado.

I think I’d have a tendency to say, “por ningún lado” and “en ninguna parte”.

And, just to complicate thing a bit, there is also the variant “en ningún sitio” : )