To add to most other answers, as you were looking for references, below you'll find some saying that you should avoid using arbitrary thresholds.
Note that there are various standardized effect sizes established by Jacob Cohen on a "small/medium/large" scale; you can find some specific examples on the wikipedia article about effect size. These thresholds or standards seem to be quoted in the documentation of some statistical software too. But if you dig a bit, you'll realize that even Cohen advised against using this "small/medium/large" standard -or any arbitrary standard, for that matter. See several of his quotes at the end of this answer, taken from the book where he established this "small/medium/large" standard.
Here are some reliable references about the problems caused by using standardized effect sizes:
Glass, G.V., B. McGaw, and M.L. Smith, in Meta-Analysis in Social Research (1981):
There is no wisdom whatsoever in attempting to associate regions of
the effect size metric with descriptive adjectives such as “small,”
“moderate,” “large,” and the like. Dissociated from a context of
decision and comparative value, there is little inherent value to an
effect size of 3.5 or .2. Depending on what benefits can be achieved
at what cost, an effect size of 2.0 might be “poor” and one of .1
might be “good.”
Kelley, K. and Preacher, K.J., On Effect Size (2012), https://doi.org/10.1037/a0028086:
Consequently, as tempting as it may be, the idea of linking universal
descriptive terms (e.g., “small,” “moderate,” or “large”) to specific
effect sizes is largely unnecessary and at times misleading (e.g.,
Baguley, 2009; Lenth, 2001; Robinson et al., 2003; B. Thompson, 2002).
Our view is that the meaningfulness of an effect is inextricably tied
to the particular area, research design, population of interest, and
research goal, and it would be inappropriate to wed effect size to
some necessarily arbitrary suggestion of substantive significance
Harrell, F., Statistical Problems to Document and to Avoid: Checklist for Authors (2020):
Many researchers use Cohen’s standardized effect sizes in planning a
study. This has the advantage of not requiring pilot data. But such
effect sizes are not biologically meaningful and may hide important
issues as discussed by Lenth. Studies should be designed on the
basis of effects that are relevant to the investigator and human
Even Jacob Cohen, who established the "small/medium/large" thresholds in his book Statistical Power Analysis for the Behavioral Sciences (1988, second edition), repeatedly and unequivocally warned against their use, and considered them as a last resort. In other words, even if you consider they might have some value (which may be debated, as seen in previous quotes), Cohen considered that you should avoid them as much as possible. For example, page 25 of his book, about the effect size $d$:
The terms "small," "medium," and "large" are relative, not only to
each other, but to the area of behavioral science or even more
particularly to the specific content and research method being
employed in any given investigation (see Sections 1.4 and 11.1). In
the face of this relativity, there is a certain risk inherent in
offering conventional operational definitions for these terms for use
in power analysis in as diverse a field of inquiry as behavioral
science. This risk is nevertheless accepted in the belief that more is
to be gained than lost by supplying a common conventional frame of
reference which is recommended for use only when no better basis for
estimating the ES index is available.
p.113, about the effect size $q$:
Again, the reader is urged to avoid the use of these conventions, if
he can, in favor of exact values provided by theory.
p.147, about the effect size $g$:
It must be reiterated, however, that a basis for positing $g$ which
comes from theory or experience should automatically take precedence
over these conventions.
p.184, about Cohen's $h$:
As before, the reader is counseled to avoid the use of these
conventions, if he can, in favor of exact values provided by theory or
experience in the specific area in which he is working.
p.224, about the effect size $\omega$:
The best guide here, as always, is the development of some sense of
magnitude ad hoc, for a particular problem or a particular field.
and so on.