Much depends on whether an individual has other criminal issues in the past. In many counties, individuals arrest for drunk in public, PC 647(f), are arrested not booked, put in a drunk tank, and then released. In others, they are booked, then the DA's office reviews the report and decides not to file charges depending on the facts. If the case was not filed and "rejected" by the DA's office, then you may be eligible to obtain a "certificate of detention only." In essence, that will change the label of "arrest" to "detention only" which may lessen any impacts if any. The best thing to do is to consult an immigration lawyer in your area.
You should consult with an attorney with your past criminal history of all arrests and convictions, if any. Just by its self, arrest for drunk in public and later release without filing of charges, should not affect your right to seek adjustment of status.
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Most likely not, as you could be on probation after a conviction and still be able to obtain a green card, depending on the nature of the crime.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.