Bigamy
A second marriage during the subsistence of a valid first marriage is void, and it is an offence. It is also a ground for divorce, and additional grounds are available to a wife where the husband has remarried.
Proof is the difficulty. Courts require proof that the second marriage was actually solemnised with the essential ceremonies: cohabitation, a photograph, or an assertion of marriage is not enough on its own. Temple records, wedding invitations, witnesses to the ceremony and any registration are what carry the case.
Adultery after Joseph Shine
The Supreme Court struck down the criminal offence of adultery in 2018. It is no longer a crime, and no prosecution lies for it.
It remains, however, a ground for divorce under the matrimonial statutes, and it can be relevant to maintenance and, in limited ways, to custody where the conduct affects the child. Proof is rarely direct; courts draw inferences from circumstances, and evidence obtained unlawfully creates more problems than it solves.
Allegations, evidence and the risk of overreach
Both allegations are damaging on the record and both are frequently pleaded on suspicion. An unproved allegation of adultery or bigamy tends to rebound: it damages credibility on everything else, and can itself be pleaded as cruelty by the other side.
Our advice is consistent. Do not confront, do not attempt surveillance that crosses into illegality, and do not put an allegation on record until the evidence supports it. Where the evidence is genuinely there, plead it precisely and prove it: see also perjury and false affidavits where the other side has lied on oath.