Christina Applegate -- Fiance Threatened on Twitter ... 'I Am Going to Kill You'

A creepy Twitter fan is threatening Christina Applegate and her fiance with terrifying messages like ... I'll "blow your f**king head off!!!" -- and LAPD has launched an investigation, TMZ has learned.

Law enforcement sources tell us Applegate's fiance, musician Martyn LeNoble, got the threatening tweets on Sept. 20th -- and according to documents filed in the case, they included: "You stole [Christina] away from me. If not for you she would be mine."

The tweet continues ... "I will not accept this. I am going to kill you!!!!!!!!!"

LAPD obtained a search warrant 6 days later -- which was served on Twitter HQ in San Fran -- and the docs list several other violent threats:

- "I'm going to bust your head in for taking her away from me!!! Be afraid, be very afraid!!"

- "You think this is a joke? I want to see how funny you think I am when I blow your f**king head off!!!"

The big problem cops are facing is ... Twitter is so damn anonymous.

According to the search warrant, LeNoble didn't recognize the Twitter profile sending the threats. Police want to access Twitter's records related to the profile ... which has been deleted.