Good question. I’m sure the poskim have addressed whether mamzerim should marry. They’re technically allowed to marry gerim or other mamzerim, but I’ve definitely heard of of people telling mamzerim they will never get married.
Of course suicide is out of the question.
Keep in mind that some sources counsel mamzerim or doubtful mamzerim to conceal their identity and marry a kosher Jew anyway (perhaps relying on the halacha that an unknown mamzer is kosher).
Also keep in mind that batei din nowadays will never in practice declare someone a mamzer, even if there is overwhelming evidence that they are. Some apikores in Israel tried her best to get her child declared a mamzer, solely to create a chilul Hashem, but the batei din refused.