Certainly in every case, you must list all of your creditors (even the ones you want to keep like a house and car and even the ones that are not dischargeable like student loans).
The filing of the bankruptcy will appear and remain on your credit report for ten (10) years, and anyone that has access to your credit report will know. While the court will need to know your employer information, the court will not inform your employer of your filing in a Chapter 7 case.
In a Chapter 13 personal reorganization there is a chance that the court may require that the plan payments come from your payroll if you work for a large employer. However, an experienced bankruptcy will be willing to ask that the court not make such a requirement if it might have a negative impact on your employment.
If you are being garnished, we will notify your payroll department to stop the garnishment immediately upon filing the case, but we will not notify anyone else without giving you a good reason.