Can I obtain a Secret Security Clearance?

was arrested for criminal mischief and public drunkeness. The charges were dropped.

They were just summary offenses (like tickets) and the charges were dropped, will I have trouble obtaining my SECRET clearance for a job i just was offered?

If this was your only brush with the law and you can mitigate it to an adjudicators satisfaction, you have a decent chance. If it was for a Top Secret, it would be a different story. Anything with alcohol triggers a HUGE red flag in the security community. There is heavy emphasis on this behavior, especially since it was only one year ago. Two years or more is preferable, but try your best to explain the situation fully. Since the charges were dropped with no courts or jail time, you might have a shot at it. Make it clear you're not a drug user or alcohol abuser, you have steady employment, out of debt and have solid references. Good luck!