OK, this should be easy. This is not a "who" question or a "what" question (both fairly obvious). Why is this happening and why is it significant?

Photograph by Andrew H. Arnott / Los Angeles Times
Marlene Dietrich is fingerprinted by Jean Armand at the Hollywood Canteen, Oct. 1, 1942.

• "Marlene Dietrich is being fingerprinted as part of the process of finally,
  after long delay, applying for American citizenship in 1939." No. This has nothing to do with her citizenship status.
• Is it because she was a German national and we were at war with Germany? No.
• She’s being fingerprinted in preparation for her first USO tour during the early days of WWII? No, but you’re on the right track. The photograph was taken in 1942 and was for something involving servicemen.
• So she could work at the Hollywood Canteen?

Exactly right. Marlene Dietrich is being fingerprinted for a photo ID issued to everyone who worked at the Hollywood Canteen. A closeup shows her signature and her card.

Here’s what the photo IDs looked like. They occasionally show up on EBay and are quite expensive.

Front

Back

The significance? Here’s the Hollywood Canteen’s photo ID of murder victim Georgette Bauerdorf, published after she was killed in 1944.

What’s important is that although some crime books claim Elizabeth Short, the Black Dahlia, worked at the Hollywood Canteen, no similar photo, fingerprints or ID card were ever found for her.  Even more evidence, along with the list of her movements prepared by the district attorney’s office, that she was only in Los Angeles for a brief time with her father from December 1942 to January 1943, and after the war, when the Hollywood Canteen had closed.

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ps. How about those eyebrows?