Criminal record is a very serious reason which can have the PR application be denied, no matter how "minor" it was. Several factors must be looked at (when it happened, what's the infraction in the home country, would it lead to a conviction in Canada too, etc.) It's an absolute must to contact a lawyer and assess such an issue before taking any step to any immigration program to Canada, including the startup visa program.
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How much does a foreign investor's past criminal record matter when applying for a startup visa in Canada?
Would the chances for a startup visa be slim if the foreign applicant had a conviction for a minor offence, like shoplifting or drink driving or a traffic violation in the home country?
It is the same rule for any program. The fact that it is a startup business class visa is not relevant. The rehabilitation rule would prevail. I recommend you do the rehab process before sending the application.