People who collect unemployment are required to complete work search activities each week they submit a claim. The number and variety of work search activities varies by state. You need to document the employers name, address, and telephone number. You are not required to have a manager sign your form. Your signature is required on the form to indicate that you authorize the information you supplied is correct and that you understand the penalties, such as prosecution and ineligibility for benefits, provided by law for making false statements for the purpose of obtaining benefits. The state reserves the right to check with any of the prospective employers you apply to so making up random places would not be a good idea.
The unemployment office wants to see that you are making an honest attempt to find employment. This means that applying to a craigslist ad with no employer information would not be considered a valid work search activity. There are a variety of work search activities you can participate in. You can apply in person, call an employer, send your resume by mail, apply online, or attend a job fair, etc. If you see an ad that does not list the employers name or address it is most likely a scam and they are not really hiring.