I asked George McNulty, and here is his answer. It partially coincides with my answer here, but is much more complete.

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I think the first result of
this kind is an immediate if unstated consequence of a result in Peter Perkins dissertation

P. Perkins, ``Decision Problems of Equational Theories of Semigroups and General Algebras'', University of California, Berkeley 1966.

In a signature with two binary operation symbols and two constant symbols, Perkins
proves (his Theorem 36) that the collection of finite sets of equations that are bases of finite algebras is undecidable. He does this by reducing the word problem on a particular finitely presented semigroup to this question. Loosely, if the word w is a consequence of the semigroup presentation, then the associated finite set of equations will be a base of a finite algebra, whereas if w is not a consequence then the free algebra on one generator in the variety based on the set of equations will be infinite. Of course, this also shows that the locally finiteness problem
is also undecidable. This part of Perkins dissertation was published as

P. Perkins, ``Unsolvable problems for equational theories'', Notre Dame Journal of Formal Logic, vol. 8 (1967) 175--185.

One of the things I did in my 1972 dissertation was to establish various extensions of Perkins work on this topic. In particular, I showed that the above result holds for any finite signature that has an operation symbol of rank at least two. I had a rather long list of properties of finite sets of equations or of the varieties based on finite sets of equations that I could prove to be undecidable, but I didn't put all the proofs even in my dissertation. By the time I came to write it up for publication, I had figured out a handful of results from which most of the undecidability results I knew would follow. I published these in

G. McNulty, ``Undecidable properties of finite sets of equations''. Journal of Symbolic Logic, vol. 41 (1976) 589-604.

You can find in that paper a long list of such properties, but local finiteness is not
on the list, while being the base of a finite algebra is. The local finiteness business
follows in the same way as it did from Perkins result.

There are only a handful of other papers that address undecidable properties of
finite sets of equations (mostly, I think, because undecidability seems to prevail---although
some result like the Adjan-Rabin Theorem is unknown). Here they are:

V.L. Murskii, ``Nondiscernible properties of finite system of identity relations'', Doklady Akademii Nauk SSSR vol. 196 (1971) 520--522.

This paper is independent of my work or Perkins work. There is a large overlap between
Murskii's findings and what is in my dissertation, although this 3 page account of Murskii's work is, of course, very terse. I don't think Muskii's work covers either being the base of a finite algebra or being the base of a locally finite variety, but it is very interesting. Murskii was the first to frame a general condition on collections of finite sets of equations that would ensure undecidability. It was Murskii's paper that spurred me
to frame other general conditions that you can find in the paper of mine above. (I also
include there a second proof of Murskii's general condition.

Douglas Smith in his 1972 Penn State dissertation found another undecidability
result. It is in

D. Smith, ``the non-recursiveness of the set of finite sets of equations whose theories are one-based.'' Notre Dame Journal of Formal Logic, vol. 13 (1972) 135--138.

Ralph McKenzie wrote

R. McKenzie, ``On spectra and the negative solution of the decision problem for identities having a nontrivial finite model.'' Journal of Symbolic
Logic, vol. 40 (1975) 186--196.

Don Pigozzi wrote

D. Pigozzi, ``Base-undecidable properties of universal varieties.''
Algebra Universalis, vol. 6 (1976), no. 2, 193–223.

Among other things, Pigozzi shows that it is undecidable whether the
variety based on a finite set of equations has the amalgamation property
or the Schreier property (subalgebras of free algebras are free).

C. Kalfa, ``Decision problems concerning properties of finite sets of equations.
Journal of Symbolic Logic, vol. 51 (1986) 79--87.

Here Cornelia Kalfa shows that the joint embedding property is undecidable, as is
whether the elementary theory of the infinite models is model complete.

The latest paper I know about is

C. O'Dunlaing, ``Undecidable questions related to Church-Rosser Thue systems.''
Theoretical Computer Science, vol. 23 (1983) 339--345.

Here it is shown that it is undecidable whether of finite set of
equations is logically equivalent of a finite confluent set of equations.

Recently Ralph Freese and some collaborators have found fairly quick algorithms
for a lot of the kind of properties above when the equations examined have certain
restricted forms.

That's about what I know about this line of research. Hope some of it is useful.

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George also wrote:

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Also I noticed the interest expressed about free spectra in the original posting. There are a lot of papers on this (and related topics). Perhaps Joel Berman is the person who knows all about it.