Immigrant visas are always immediately available for the spouses of U.S. citizens. There are two ways in which your immigrant spouse can obtain his/her green card—either through (1) Adjustment of Status in the United States, or (2) Consular Processing at a foreign consulate. Each of these processes achieves the same result, but they are very different procedurally.
The adjustment of status application is filed in the United States, and therefore is only to be used if your foreign spouse had a valid visa when he/she entered the U.S. Even if your spouse’s status has since expired, he/she may still be able to adjust status in the United States. If you married outside the United States and/or your spouse does not currently have a visa to come to the U.S., your spouse would need to consular process. Please note that if your foreign spouse ever entered the U.S. unlawfully, he/she may be need to pursue a Hardship Waiver before being able to obtain Lawful Permanent Residence.
The main issue when it comes to obtaining a green card through marriage is demonstrating that the marriage relationship is bona fide. This usually requires extensive documentation throughout the application process as well as a detailed interview of both parties by an immigration officer.