Do nofollow backlinks influence ranking?

Selected answers from the Dumb SEO Questions Facebook & G+ community.

• 
  Jay Le Boutillier: Is nofollow back-links influence today's Google SERP
• 
  Saurabh Rawat: There is no proof on the net but my study says.. indirectly nofollow back-links can ;influence your ranking and ratio between nofollow and dofollow links can be matter. Lots of do follow links with compared to nofollow links are harmful for site so no follow links are also necessary.
  No follow links on quality sites and social media sites can improve your website quality and popularity.
  But imortant thing is the dofollow is more beneficial as compared to nofollow for ranking if it is relevant. ;
• 
  ?ukasz Rogala: Care about traffic you can get instead of do or nofollow thing. ;
• 
  Jann Grond: Google doesn't want to count those links, they want editorially earned links!
• 
  Saurabh Rawat: Yes.. Make relationship not links, Google more likes quality of links not quantity of links.
  Content Marketing is the best solution.
• 
  Tim Capper: Forget about links being followed or not.
  
  If you could get a link in the new York times that would send thousands of visitors and potential customers. ... would you say no thanks. ... its no follow 
• 
  Saurabh Rawat: Yes, +Tim Capper ;has gave a good example..
• 
  ?ukasz Rogala: One of my friends in Poland have a case where he mostly get nofollow links and after a great on-site optimization he ranked for a heavy Polish search query. I think it was about 30% dofollow to 70% nofollow links but nofollow links were from trusted and well ranking sources in Poland (some top news websites etc.)
  
  So in general even if they do not count it's good to have them - especially if they bring traffic and conversions. :)
• 
  Saurabh Rawat: Final Touch - Google Webmaster Guideline says..If you are creating links on untrusted sites or creating paid links than use nofollow that's mean Google does not give any penalty for nofollow links.. but nofollow links are beneficial for traffic and brand awareness. so make nofollow on relevant sites and popular sites (Wikipedia) for traffic. nofollow only for ranking it is bad idea.
  https://support.google.com/webmasters/answer/96569?hl=en
• 
  Jay Le Boutillier: Thanks to all for Sharing your Ideas and review with me