It will depend largely on exactly what morph issue you are talking about and whether it is incomplete dominant or recessive. Generally speaking, the animal that shows the visual morph will/may display the issue, while animals that don’t visually display the morph are either het for the gene (and may carry the issue) or don’t have the gene at all, in which case, wouldn’t carry.
In the case of neuro morphs (champagne, woma, HGW, Spotnose, spider) the phenotype is directly linked with the neuro issues, and as such it is impossible to breed out. Any visual neuro morph animal will have the neurological syndrome, whether we can see a wobble visually or not. An animal that is not a phenotypic example does not carry the gene, and therefore does not have or carry the issue.
In the case of things like super black pastel/super cinnamon kinking and duck billing, animals carrying one copy of the gene display no issues and only animals in the homozygous form have a higher likelihood of those issues.
In the case of fertility issues (caramel albino and desert), the phenotypical animal will be at high risk for the issue and in the case of the recessive, a het animal (carries only one gene) will carry the gene that causes the risk.