Speaking to presenter Kirsty Wark, an emotional Allen described how she was stalked by Alex Gray intermittently for seven years, although she was unaware of his identity for much of that time.
She tells how she gave the police letters which he had sent to her, and how police refused her request for a photograph of her stalker so she could have some idea of what he looked like.
Eventually she was shown one, albeit briefly. In October 2015, he finally broke into her house while she and her children were in bed.
The Metropolitan Police said it took stalking "extremely seriously" and sought to put victims at the heart of any inquiry.
You can hear Allen's abridged story here. Below, we print more extracts from her 25-minute interview with Wark.
Early in 2009 I played a concert at Koko which is a relatively small venue in Camden. I was on stage singing and during the concert I saw - often people hold banners up or messages for me - and I saw this banner that said "I wrote The Fear, where's my money" and I immediately knew that it was him and that was quite terrifying because I didn't know if he had a weapon on him ... Also I was the only person in the building that knew who he was and that was a really isolating moment for me, knowing that he knew that I knew but nobody else knew. I finished the song and then ran to the side of the stage where my assistant, who had been there, was on the side of the stage and said "Vicky, that's him. Can you call the police."
They came and installed a panic alarm at my flat. (She had no cause to use it). They came back and took it after six months - I didn't think there was much point in my objections. I knew what the answer would be - which was "it's been six months, there hasn't been any contact, we're going to take the alarms away now".
I believe it was in 2012. They [the police] said "no" at first and we pushed them for a photograph and they came over and showed us a photograph and then took it away.
I assumed that they would call me into the police station and there would be a line up but they didn't. I knew the outcome of the bail hearing might be that he's out on bail so I wanted to be there for that bail hearing on the off-chance he was released. I didn't quite trust the system to call me and find me if he was released so I wanted to make sure for myself that a) it was the right person and b) that he wasn't going to be released and, if he was, that I knew about it straight away.
I kept asking the police where are these letters, were they adding them to the indictment because, as far as I was concerned, that built our case for stalking but it didn't really seem like they were very interested in making it a stalking case. I assumed they would be adding them to the investigation but it turns out the letters were destroyed. I did have to take it upon myself to get the stalking charge added, which cost me a lot of money.
I felt like every time I went to court it provoked something in him that alerted the judge to how serious this thing was because I didn't really feel like the police or the Crown Prosecution Service were taking that seriously so I kept wanting to be there so that the judge or whoever was dealing with him could see what his feelings towards me were and deal with it appropriately. Police weren't aware that I was going to these court sessions - they didn't tell me about them, I found out about them myself.
I have felt more closure speaking to the Observer and speaking to you about this subject than I have about this whole investigation. I feel more strength now than I did on the day he was convicted. I feel like it's my duty to do this. There are women suffering who are terrified. It's life changing. It's life stealing is what it is. It's living in fear and it's really important to me to help other people in that position.
Earlier this month, Alex Gray, 30, from Perth, was convicted of harassment and burglary. He is awaiting sentencing.
The singer Lily Allen has spoken to BBC Newsnight about her experience of being stalked and how she believed the police response was to "victim-shame".