This is a variation of the coupon collector problem considered in section 5 of Some new aspects of the coupon collector’s problem (2003).
The $t$ singleton sides (sides which appear once) can be chosen as an ordered sequence in $t! {s\choose t}$ ways; this sequence can appear among the $n$ rolls in ${n\choose t}$ ways and the remaining $n-t$ rolls constitute an ordered partition of $n-t$ elements into $s-t$ classes, no class having fewer than two elements, which can be chosen in $(s-t)!\left\{{n-t}\choose{s-t}\right\}_2$ ways. Multiply these together and divide by $s^n$, the number of $n$ sequences, to obtain the result.