This issue was addressed a while back in the Journal of Halachah and
Contemporary Society.
I can't recall the details of the discussion, but it went something like this.
Basically, the big problem with mamzerus is that any child sired via a
Jewish mother by a mamzer is born a mamzer. This applies whether the mother
is a convert or born of a Jewish mother.
So, theoretically, the only way to break the mamzerus chain, is to have a
child with a non-Jew and then have the child convert. Since a ger's legal
status is completely unrelated to status of his or her father, the father's
mamzer status would not transfer to the biological child.
The catch is that any relations with a non-jewish woman would count as
z'nus, at least, and are forbidden.
The article therefore addresses a theoretical way to remove mamzerus, based
on this principle, in a legal fashion.
Essentially, a Jewish servant is permitted to have relations with a
non-Jewish maidservant (with the children born thereby, becoming part of
the master's household). Once the children and maid-servant leave the
master's household, they become Jews in the full and normal sense. Thus,
the children born of this special permitted relationship with a non-Jewish
woman, would be considered like any other convert, as per the above principle.
In such a fashion, a mamzer could have permissible relations which result
in a child who could convert and become a full Jew, without the problem of
mamzerus. Of course, the major catch in today's society is that it is not
even clear if we permitted to have avdus(servants) any more...but that is a
separate discussion...
I'm sure I've made a couple of errors here, so please feel free to correct
them. It's an unusual issue to be sure...
Eliezer Gamerman