Blocking semiovals of Type (1,M+1,N+1)

We consider the existence of blocking semiovals in finite projective planes which have intersection sizes 1, m+1 or n+1 with the lines of the plane for $1 leq m < n$. For those prime powers $q leq 1024$, in almost all cases, we are able to show that, apart from a trivial example, no such blocking semioval exists in a projective plane of order q. We are also able to prove, for general q, that if q2+q+1 is a prime or three times a prime, then only the same trivial example can exist in a projective plane of order q.