  We give a classification of four-circulant singly even self-dual $[60,30,d]$
codes for $d=10$ and $12$. These codes are used to construct extremal singly
even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal
singly even self-dual code was previously known to exist. From extremal singly
even self-dual $[60,30,12]$ codes, we also construct extremal singly even
self-dual $[58,29,10]$ codes with weight enumerator for which no extremal
singly even self-dual code was previously known to exist. Finally, we give some
restriction on the possible weight enumerators of certain singly even self-dual
codes with shadow of minimum weight $1$.
