Yes. You may file a green card application if you have criminal record. HOWEVER, Criminal Record may be an obstacle and can affect your eligibility to adjust status in the United States and get a green card. Furthermore, you may also get arrested and be placed in Removal Proceedings.
As part of the process of adjusting your status in the United States and becoming a green card holder, the USCIS requires that you be fingerprinted. The fingerprint process allows the USCIS to check your background and if you have criminal record, arrest or convictions it will come up in the background check and your green card application will be at risk and may be denied. In some case you may be eligible to apply for a Waiver of Criminal Conviction. On the other hand, the USCIS may deny your green card application and place you in Removal Proceedings.
Therefore, if you have ever been arrested, cited or stopped by the police either in the United States or anywhere else in the world you should consult with a US Immigration Lawyer before you file your green card application.