Police are investigating allegations against a man from Braintree in Essex of criminal communication through social media.
They relate to "inappropriate and offensive" Twitter posts about the Boston bombing, the Hillsborough disaster and James Bulger's murder.
An Essex police spokesman said the complaints were made on Thursday by a person living in Southport, Merseyside.
He said no arrests had been made and inquiries were continuing.
The investigation follows controversy on Twitter relating to tweets apparently made by an anonymous user and blogger using the alias Old Holborn.
The name, address and telephone numbers of a man purported to be Old Holborn were subsequently posted online.