What does gene nonspecific and allele nonspecific mean, formally?
Cross the suppressor to many different genes and many different alleles of each gene, then look for suppression/rescue. Gene nonspecific means that the suppressor will suppress the mutant phenotype when crossed to several different genes. This occurs in the case of an informational suppressor, which suppresses a certain specific kind of allele (e.g. amber nonsense mutation) which can arise in any gene nonspecific. Allele nonspecific means that the suppressor will suppress many different alleles, usually of the same gene. This occurs in the case of a bypass suppressor, which makes up for the function of a gene involved in the same process, no matter what kind of mutation affected that gene.