Apparition of scale invariance 
When did "scale invariance" started to be seen as an important concept
  in the theory of phase transition?

Phase transition and critical points started to be investigated in earnest in the middle of the 19th century, with the study of critical opalescence  (Andrew/Faraday) : at the critical point of a liquid, the liquid start to get cloudy/diffuse, a phenomenon widely attributed to the scale invariance of the medium nowadays. I like to pinpoint when scale invariance was first observed, and when it started to be seen as something important in the theory of phase transition.  
It seems that a lot of what was needed to introduce scale invariance in physics was already here at the start of the 19th century, with brownian motion and fractals being investigated. Marian Smoluchowski was the first (at the beginning of the 20th century) to guess that opalescence was due to large fluctuations in the fluid (but that's not "scale invariance"). 
Conformal invariance (a generalisation of scale invariance), started to get applied to phase transition in the 70's. So that's an upper boundary I guess.
A google ngram seems to indicate that "scale invariance" only became a thing when conformal invariance started. But one of A. Migdal anecdote seems to indicate that scale invariance was already seen as "old news" in the beginning of the 70's (pdf, page 34-35).
Sorry if the "history of science" question is out of bounds for this stackexchange. 
 A: According to this paper by Michael Fisher, one of the pioneers in this area, the notion of scale invariance, in the context of the critical point, appeared in the mid-1960s. Let me quote two passages from the paper; much more information can be found the paper, of course.
From the introduction of the paper:

Whence came RGT? This is a good question to start with. I will try to
  respond, sketching the foundations of RGT in the critical exponent
relations and crucial scaling concepts of Leo P. Kadanoff, Benjamin
  Widom and myself developed in 1963– 1966 — among, of course, other
  important workers, particularly Cyril Domb and his group at King’s
  College London, of which, originally, I was a member, George A. Baker,
  Jr., whose introduction of Padé approximant techniques proved so
  fruitful in gaining quantitative knowledge, and Valery L. Pokrovskii
  and A. Z. Patashinskii in the Soviet Union who were, perhaps, the
  first to bring field-theoretic perspectives to bear.

and from Section 3:

The 12 months from late-1965 through 1966 saw the clear formulation of
  scaling for the thermodynamic properties in the critical region and
  the fuller appreciation of scaling for the correlation functions.
  One may highlight Widom’s (1965) approach since it was the most direct
  and phenomenological — a bold, new thermodynamic hypothesis was
  advanced by generalizing a particular feature of the classical
  theories. But Domb and Hunter (1965) reached essentially the same
  conclusion for the thermodynamics based on analytic and
  series-expansion considerations, as did Patashinskii and Pokrovskii
  (1966) using a more microscopic formulation that brought out the
  relations to the full set of correlation functions (of all orders).
  Kadanoff (1966), however, derived scaling by introducing a completely
  new concept, namely, the mapping of a critical or near-critical system
  onto itself by a reduction in the effective number of degrees of
  freedom.

Obviously, scaling concepts in other settings can be found earlier (for example, in Kolmogorov's work on turbulence in the early 1940s).
