I am a United States citizen that has been married to an illegal immigrant for eleven months now. He entered the country illegaly, has been here for almost 5 years. My husband has never worked with someone else's documents, he has never been in trouble with the law, not even a ticket, nor has he ever been fingerprinted. He works for a company that moves heavy machinery, and does his taxes every year.
I forgot to add that he pays child support.
Unfortunately, the fact that your husband entered illegally (not necessarily that he is not in legal status now) is a big problem here.
With legal entry, even someone out of status can still get a green card in the U.S. without leaving based on marriage to a citizen.
But if the person entered illegally, they would need to leave the U.S.and process through the U.S. consulate in their home country - and the act of leaving after one year of "unlawful presence" makes him subject to a ten-year bar to coming back.
There is a waiver of this bar available based on extreme hardship to a U.S. Citizen spouse or parent - you should speak with an attorney to see if this is a viable possibility.