Thirty days. Hire an experienced ticket attorney to handle this for you.
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Sorry, but I'm afraid you definitely do NOT have 30 days to appeal any Class C conviction. You've only got ten days after the judgment's entered to appeal, and you appeal it by posting twice the amount of the fine and costs you were assessed with the clerk of the court the case was in (usually it's twice that, technically that's the minimum and it can be more if the judge wants it to be). While that part of it obviously wasn't what you were probably hoping to hear, the good news is that in the vast majority of cases, an attorney will be able to either get your case dismissed altogether or at least get your deferred disposition once it's appealed up to county court, and either of those will let you later get an expunction.
Here's the Code of Criminal Procedure statute section regarding the ten days to appeal:
Art. 45.0426 : Filing Bond Perfects Appeal
(a) When the appeal bond has been filed with the justice or judge who tried the case not later than the 10th day after the date the judgment was entered, the appeal in such case shall be held to be perfected.
(b) If an appeal bond is not timely filed, the appellate court does not have jurisdiction over the case and shall remand the case to the justice or municipal court for execution of the sentence.
For convictions in municipal courts, you have 1 day to file a motion for new trial In the trial court and it must be heard within 10 days.