Does anyone use $\subset$ for proper subset anymore? I belong the the group of people who still write (not necessarily proper) subset as $\subseteq$ to avoid any confusion with proper subset, which I notate $\subsetneq$; I usually do not use $\subset$ at all. But now that I think about it, I have not seen anyone use $\subset$ for proper subset in any post-1960s textbooks. Is it still used for that at all, or can that use of the $\subset$ symbol be considered archaic?
 A: It is hard to say what is common or archaic use of a symbol, because it would require studying a large number of publications.
The use of $\subset$ for proper subset and $\subseteq$ for subset is obviously motivated by analogy with $<$ and $\le$. On the other hand, using $\subsetneq$ for proper subset and $\subset$ for subset is motivated by the fact that proper subset is a concept which is seldom needed in actual mathematical context. (I cannot easily create a meaningful example where proper subset is required.)
At my university (Budapest), only $\subset$ was ever used and always meant subset. No one ever used $\subseteq$ (nor $\subsetneq$), because the concept proper subset was seldom used, and the simpler form, $\subset$ had a better use for the more common meaning subset (a kind of economical choice of symbol). In fact, these symbols had a certain "childish" or at least "high schoolish" look. On the rare occasion that proper subset was needed, one would write $X \subset Y, X \ne Y$, in order to emphasize this constraint.
Below I collected some of the global resources that may be useful in deciding one way or other:
ISO
ISO has a standard for math symbols, but it is targeted towards natural sciences and technology rather than mathematics itself.
The standard is behind a paywall, so only the Wikipedia page can be linked here: ISO 80000-2:2009. An eralier, obsolete standard (which nevertheless is very similar) is available on Wikipeda: ISO 31-11:1992.

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*Item 2.5.7 defines $B \subseteq A$ as "$B$ is included in $A$, $B$ is a subset of $A$".

*Item 2.5.8 defines $B \subset A$ as "$B$ is properly included in $A$, $B$ is a proper subset of $A$".

However; even the ISO document contains a remark for 2.5.7 and 2.5.8 that defines an alternative: $\subset$ is subset and then $\subsetneq$ must be used for proper subset.
Wikipedia
The Wikipedia article on math symbols takes the same approach, preferring the pair $\subseteq$ / $\subset$, but mentioning $\subset$ / $\subsetneq$ as well.
Unicode character names
Unicode defines these operators in the Mathematical Operators block and uses the following names:

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*U+2282 $\subset$ SUBSET OF

*U+2286 $\subseteq$ SUBSET OF OR EQUAL TO

*U+228A $\subsetneq$ SUBSET OF WITH NOT EQUAL TO

This seems to prefer the $\subset$ / $\subsetneq$ pair, but is not decisive (ambiguity is very common with Unicode names, unfortunately).
HTML entities
In the official HTML entity list (renamed to "named character reference" since HTML 5):

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*$\subset$ is &subset;

*$\subseteq$ is &sube;

*$\subsetneq$ is &subne;
Again, this seems to favor the $\subset$ / $\subsetneq$ pair.
Summary
Where I come from, professional mathematicians prefer $\subset$ for subset. Proper subset is virtually not used. If a single character were still required for that case, it would be $\subsetneq$. But better avoid that as well.
In education and applied sciences one can see $\subseteq$ for subset more often.
There are also some mentions of $\subset$ meaning proper subset, but I have never seen an actual example where it was used like that. In order to avoid confusion, one could (in fact, should) use $\subsetneq$ for proper subset even if $\subseteq$ is used for subset.
Bottom line: use of $\subset$ meaning proper subset is ill-advised in any case because it creates confusion. Use either

*

*$\subset$ for subset and (if absolutely necessary) $\subsetneq$ for proper subset (my preference), or

*$\subseteq$ for subset and $\subsetneq$ for proper subset (totally unambiguous choice).

