The appearance of impropriety is a phrase referring to a situation which to a layperson without knowledge of the specific circumstances might seem to raise ethics questions. For instance, although a person might regularly and reliably collect money for her employer in her personal wallet and later give it to her employer, her putting it in her personal wallet may appear improper and give rise to suspicion, etc. It is common practice in the business and legal communities to avoid even the appearance of impropriety.
1 Thessalonians 5:22, in the King James Version of the Bible, says "Abstain from all appearance of evil."