What court has the authority to grant and issue writs of habeas corpus in criminal law matters?

The Court of Criminal Appeals is the correct choice because it is specifically designated to handle appeals involving criminal matters. In many jurisdictions, including states in the U.S., this court has the authority to grant writs of habeas corpus, which are legal instruments used to challenge the legality of a person's detention or imprisonment.

Habeas corpus serves as a vital safeguard against wrongful imprisonment, ensuring that a court evaluates whether the detaining authority has lawful grounds for holding an individual. As cases often pertain to criminal law issues, the Court of Criminal Appeals is especially focused on interpreting criminal statutes and procedures, making it the appropriate venue for such writs.

The other courts mentioned typically do not have the same jurisdiction or focus. For instance, the Supreme Court usually handles issues of broader significance or constitutional matters but does not operate as a trial court for criminal cases. District Courts and Municipal Courts generally have limited authority and do not deal exclusively with the complexities associated with habeas corpus, which often require a deeper consideration of criminal law and appeals.