Sócrates most likely does not ground the singleton of Sócrates

1. First of all, because the singleton of Socrates does not exist.
   
   
1. Singletons of concrete objects are not bona fide mathematical objects. Set theory is a well established and perhaps fundamental mathematical theory and so its objects are as well established part of our ontology as any other mathematical objects, but this mathematical theory contains no singleton of Socrates, it only contains pure sets and so it so its only singletons are singletons of other sets.
2. Carlos Romero reminds me that ZFC-U is a bona fide mathematical theory [Thanks Carlos!!], which is completely true; however I am not sure that Socrates is an ur-element of the sort that are the proper objects of ZFU. I suspect there is a distinction here between being an ur-element and playing the role of being an ur-element – to borrow a useful distinction from Stewart Shapiro.
3. We cannot get  singleton of Socrates via some sort of indispensability argument from the application of set theory with concrete ur-elements. Yes, it is common to find singletons of this sort in formal semantics (where they are usually used to model the denotation of single terms), but – besides the issues raised in the previous point – the consensus there is that they are neither indispensable nor the denotation of any expression in natural language.
   [For a brief, but detailed discussion of this issue see section 3 of Winter and Sche's entry "Plurals" on the Handbook of Contemporary Semantic Theor (1995), where they quote Link (1998) statement that “for practical reasons (for instance, because people are ‘used to it’) we could stick to the [use of singletons to model the reference of singular terms] as long as we don’t forget it is only a model” (p. 64) as an example of the current consensus regarding impure sets. The situation in other areas of applied mathematics are not different. Sets are too complex and well-behaved to be indispensable in our theories of non-mathematical reality.

Of course, logics with well-defined interpretations will always be caricatures of our actual thought and our actual language ... Science makes models, and models are caricatures. And that is fine. We do not have to believe in them. They are just food for further thought. (Scha 2013)] 

 

1. Second, even if sets like the singleton of Socrates existed, there is little reason to think that they would be grounded in their unique element.
1. In general, philosophers of mathematics do not take membership to be some sort of grounding relation. Stewart Shapiro has even gone as far as explicitly argue that it is not a grounding relation.
2. Yes, it is possible to offer an account of membership as grounding, but then it becomes a very sui-generis sort of grounding (i.e. it is not necessarily well grounded, since AFA is a fairly well accepted mathematical theory)
3. Your best bet is to endorse some metaphysical form of the iterative conception of set developed by Scottus, Wang, Potter, etc. where – to quote Boolos – "elements of a set are "prior to" it." (Boolos 1971: 2016) Foster, Potter and other have endorsed this way of arguing that sets are at least partially grounded in their members. However, there is still ample debate as to whether, even within the iterative conception of sets, membership is some form of grounding. I am open to accept it is, but I am genuinely skeptical.