Is the set of codes of Deterministic Finite-State Automata a regular language?
This answer is, in a sense, a completely cheating approach, but it is indeed possible to encode all DFAs as strings.

We can write out a DFA by writing out its transition table.  We can write out the transition table using just 0s and 1s as follows: first, write out a number of 1s equal to the number of states, then a 0.  Then, write out a number of 1s equal to the number of symbols in the alphabet, then a zero.  Then, write out each row of the transition table by writing out each entry as a number of 1s indicating which state should be transitioned into, then a 1.

Now, this particular encoding of a DFA is not regular.  However, what we can do is the following.  Consider the set of all such encodings.  We can then order them in length-lex order, and then can number the DFAs produced this way 0, 1, 2, 3, 4, ..., etc. based on their ordering.  In this case, we now have a bijection between $\mathbb{N}$ and the set of all DFAs.  From there, we can then consider the regular language consisting of all natural numbers written out in binary.  This set is definitely regular; here's a regular expression for it:


  0 | 1(0|1)*


So we now have a regular language consisting of encodings of DFAs.  The encoding is not at all easy to work with - you'd have to start listing off all encodings of DFAs until you found the one you were looking for - but mathematically it is well-defined.