The Euthyphro Dilemma is typically considered to be a problem for divine command theories of moral properties. The dilemma in this guise usually goes like this: Either X is good/right because God commands it, or God commands X because it is good/right. The dilemma afflicts versions of divine command theory that take the good to be prior to the right as well as versions that take the right to be prior to the good. While the dilemma is definitely an issue for divine command theories, it is not a special problem for them. The Euthyphro Dilemma can actually be raised against any theory that aims to account for something general in terms of something particular.
The Euthyphro Dilemma will be a problem for any theory that attempts to account for the general in terms of the particular. For example, exemplar nominalism has to deal with the dilemma because it attempts to account for the appearance of commonly had properties (the general) in terms of an exemplar (the particular). To account for the seemingly common property of being red, the exemplar nominalist will pick out an exemplar particular that is red and then account for commonality by introducing a resemblance relation. So, something is red if and only if it resembles a or the red exemplar. This account has to deal with the Euthyphro Dilemma, though. Either X is red because it resembles the exemplar or X resembles the exemplar because it is red. The nominalist will opt for the former horn, since the latter is to introduce universals or at least tropes.
Ideal observer theories also have to deal with the Euthyphro Dilemma. The structural similarities between these theories and divine command theories will give rise to the dilemma, because both kinds of theories attempt to account for moral properties in terms of a particular. In the case of divine command theories, the particular is God, but for ideal observer theories, the particular is usually a hypothetical, idealized member of the moral community. Call the ideal observer, “Jeffrey”. Either X is good/right because Jeffrey approves of it, or Jeffrey approves of it because X is good/right. Like the theistic dilemma, this secularized dilemma arises because the Jeffrey is a particular, and he is supposed to account for something general, which is the good and the right in this case (and the bad and the wrong).
Trying to account for moral properties in terms of Jeffrey isn’t the only way the Euthyphro Dilemma can manifest itself as a problem for ideal observer theories. If we attempt to account for aesthetic properties in terms of Jeffrey, the same dilemma arises. Either X is beautiful because Jeffrey thinks so, or Jeffrey thinks so because X is beautiful. If we try using Jeffrey to account for cognitive/epistemic goods, the same dilemma will also arise. The same thing goes for attempting to account for truth in terms of a cognitive community, which is itself a very large particular.
So, it seems to me that the common thread running through these manifestations of Euthyphro is that each theory attempts to account for something general in terms of a particular. One issue that I have not yet explored is if it is the concreteness of the particular that raises the issue, or if abstract particulars like numbers could also be problematic when used to account for the general. Let me know what you think of my analysis of the Euthyphro Dilemma in the comments below.