From what I understand, elements of a ring which are not regular are singular. So how is it possible that in the set of even numbers, $2$ is neither regular nor singular?
Edit: To clarify, an element $x$ of a ring is called regular if it has an inverse. Otherwise, it is said to be singular.
I found a reference on this topic: here
This book talks about regular and singular ring elements, and it says, "An element which is not regular is called singular". However, this author makes it clear from the start that these definitions apply to "An element $a$ in a ring $R$ with multiplicative identity".
The set of even numbers, by which I presume you mean $2\Bbb Z$, is not a ring with identity, so we're not in the right context to talk about regular vs. singular elements, at least according to the above source.