Comments about the first minister were posted in response to a Shetland Time article about her trip to the isles earlier this week. The paper has removed the post.
Similar remarks were posted on Shetland News social media pages following their coverage of Ms Sturgeon's visit.
A by-election will be held for the Shetland constituency on August 29 to elect a replacement for Liberal Democrat MSP Tavish Scott.
Mr Scott has resigned to take a post at Scottish Rugby. He represented the isles at Holyrood for more than 20 years.
In the wake of the social media posts about the first minister, Liberal Democrat candidate Beatrice Wishart said: "Sexism is utterly unacceptable and all too often turns women away from public life.
"We have a duty to call it out and I'm heartened my fellow candidates and I are united in that."
Tom Willis, the SNP's candidate for the by-election, said: "No woman should have to suffer sexist abuse online, or anywhere else.
"Gender equality transcends party boundaries and it is encouraging to see online abuse being challenged collectively across party lines. I hope this is the last time we see abusive comments like these in Shetland."
Labour candidate Johan Adamson said there is no place for "hatred or violence" towards anybody. She added: "There are three female candidates in this by-election and many women in politics who deserve respect, not least for being brave enough to stand in a world of male privilege."
Debra Nicolson, standing to represent the Scottish Greens, added that she has also been subjected to "this type of behaviour" herself.
"This misogynistic behaviour has to be called out and stopped in its tracks. It's the sort of thing that sops women getting involved in politics. It's just not acceptable."