(Throughout, triviality is taken to be in the sense of the first meaning given by the answer by 5xum to the question What defines "triviality"?)
There is, of course, no strict definition of trivial cases in mathematics, and as far as I know none for degenerate cases either. However, people seem to have some sort of intuitive sense for cases of triviality, e.g. the set of the identity element, the factors of a prime number, the indiscrete topology etc. Examples of degenerate cases include the circle of radius zero, or perhaps a counting system with radix 0, I would call these degenerate.
The case of the empty set seems to me degenerate, but is often cited as an example of a trivial case. Not that the two have to be mutually exclusive, but what then is the relationship between the two? Triviality occurs in cases which can be in a sense "taken for granted" or are explicitly stated in the definition of the objects in question, and degeneracy occurs in cases which are "useless" to what we are studying, and I could fairly easily convince myself that every example of degeneracy is also an example of triviality (I can't think of any examples where I would call something degenerate but not concede that it was also pretty trivial).
Is this a fair conclusion? Is calling a case degenerate stronger than calling it trivial in a meaningful (if not explicitly definable) way?