Instagram won’t reveal the address to you / your friend. They might be able to assist the police.
However by the same measure if the Instagram account is anonymous it seems a fair bet it would be registered with a similarly anonymous email address. Its unlikely to be own_name@own-domain.com
Really its a police issue if simply blocking the account in question isn’t sufficient (all the network can do beyond you blocking the account yourself is remove the content or close the malicious account. That doesn’t prevent another account being opened though.
The most import thing to do is keep a record – screen shot the messages even if you are deleting / blocking them or asking for them to be removed. A malicious message is one kind of offences. More that one is another.
If a person sends you threatening, abusive or offensive messages via Facebook, Twitter or any other social networking site, they could be committing an offence.
The most relevant offences are ‘harassment’ and ‘malicious communications’. For harassment to be committed, there must have been a clear ‘course of conduct’. That is, two or more related occurrences. The messages do not necessarily have to be violent in nature, but would need to have caused some alarm or distress.
If there has only been a single communication, it’s unlikely it would qualify as harassment, but could be considered a malicious communication. For such an offence to be committed, a message must be sent to another person, or sent via a public communications network, that is indecent, grossly offensive, obscene, threatening or menacing.
How to report it
You can report either harassment or malicious communications online or by calling us on 101.
We will be able to confirm whether an offence has been committed, based on the full facts and your individual situation. In order to assist us with our investigation you must not respond to the message as it may encourage the sender and make the situation worse.
If possible, take a screenshot of the message so if it gets deleted later there will still be a record of what was said.
Contacting the social networks directly
It would also be worth reporting this to the social network that was used. They will have processes in place for such situations, and may be able to simply remove the content and even close down the person’s account.