Currently, the existence of an extremal singly even self-dual code of length $24k+10$ is unknown for all nonnegative integers $k$. In this note, we study singly even self-dual $[24k+10,12k+5,4k+2]$ codes. We give some restrictions on the possible weight enumerators of singly even self-dual $[24k+10,12k+5,4k+2]$ codes with shadows of minimum weight at least $5$ for $k=2,3,4,5$. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual $[82,41,14]$ code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual $[82,41,14]$ codes with weight enumerator for which no singly even self-dual code was previously known to exist.