Zafreen Khadam, 32, is alleged to have posted links to execution videos and guides for would-be terrorists on Twitter and Whatsapp under names such as 'PrincessKuffar'.
Prosecutors say the posts "glorified" IS and "encouraged terrorism".
Miss Khadam, of Vincent Road, Sheffield, denies the charges.
Simon Davis, prosecuting, told Sheffield Crown Court she described Kuwaiti-born Mohammed Emwazi - nicknamed "Jihadi John" - as "kind of scary" but said she "would marry him".
In another message, he said, Miss Khadam declared she wanted "to live the IS life" and "supports IS and supports the mujahideen".
"The prosecution case is that this defendant was disseminating terrorist information or propaganda, or spreading the word," Mr Davis said.
"Our case is that her intention was to encourage people to commit, prepare or instigate acts of terror."
Jurors heard Miss Khadam shared the links using 14 different accounts, after several were closed down by Twitter.
The description for one Twitter profile shown to the jury, 'PrincessISback6', read: "Love Nuttella and kittens. I hate coconuts more than kufr. Used to date but then I found Allah. Pro Jihadi John's left hand".
Among the links shared were videos of Kurdish fighters being executed, seven issues of "terrorist publication" Dabiq and a speech entitled 'What is Terrorism?', the court heard.
The jury heard Miss Khadam had confirmed in interview that links to the magazines were posted by her.
Mr Davis said Miss Khadam had also provided a link to a document called Jihadi John and the Right to be Violent, and another which gave tips on how to hide from the authorities.
He said: "This is effectively a guide for a would-be terrorist to evade detection from the law enforcement agencies, and she favourited that."
Miss Khadam denies ten charges of disseminating terrorist publications between February and March 2015.
The trial continues.

Cuadrado, 28, shone in Serie A and the 2014 World Cup before joining the Blues for £23.3m from Fiorentina in February 2015, but failed to score in 13 games.
He was loaned to Juventus last season, scoring four goals in 28 appearances to help them to the league and cup double.
"I wanted him when I was Juventus coach. He will come back very soon and train and play with us," said Conte.
"Now, he's a Chelsea player and he'll stay with us. If something changes you'll know."
The arrival of Pedro at Chelsea from Barcelona last August saw Cuadrado join Conte's former club Juventus on loan five days later.
As well as playing a significant part in Juventus' successful season, he also helped Colombia to a third-place finish at the Copa America this summer.
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