"This is a new law so naturally there are certain unclear situations and grey areas that need to be ironed out," Manfred Reinthaler from Vienna police told public radio.
"At the same time there is no legal precedence."
Austria is the latest European country to ban the wearing of the full Islamic veil, known as the burqa or the niqab, in public places. The law came into force on October 1st.
But in order to avoid being sued for discrimination, the government outlawed at the same time any item of clothing that covers the face.
Government guidelines set out a number of exceptions including masks and disguises at cultural events, work wear such as medical masks, and scarves in cold weather.
But the shark costume case -- a mascot for the McShark computer chain -- and officers reportedly stopping a girl cycling in a scarf show that some police remain bewildered.
"Lawmakers did not set out the temperature (when a scarf could be worn)," said interior ministry spokesman Karl-Heinz Grundböck.
There is no central register so the total number of fines of up to €150 ($177) levied so far is unclear, Grundböck added.
The ban on the full-face veil, which remains a rare sight in Austria, was seen as the latest effort by the two governing centrist parties to halt a rise in support for the anti-immigration Freedom Party.
Polls suggest that the far-right party will garner around 25 percent support in elections on Sunday and may become junior coalition partners to Sebastian Kurz's conservatives.