I have a few things that nobody believes. I did the whole "waking up at an exact time in the night." thing for a while as a teen, it was really odd.
There was supposedly an earthquake in my home town that, IIRC, had something to do with the nuclear power station at Bradwell years ago. It happened in the early hours of the morning and local news went on about it for weeks. Loads of locals claimed that they were shaken out of their beds, could hear buildings moving and so on and so forth. I remember a few of them quoted lived in areas further from the site anda couple lived in roads right next to mine. I was up at that time talking to friends in America. Never felt a thing 
I was told I must have gone to bed earlier than I remembered. I was up until 6am, this happened between 1am and 3am supposedly!
When I was a young teen I was home alone, upstairs in my room. I heard somebody just walking around the house. Thought nothing of it before realising that I was home alone. I can't remember exactly what happened in the time between but I clearly heard somebody in the house. I grabbed a badminton racket from somewhere, crept downstairs and as I was checking out the back of the house I heard somebody run across the hallway and into one of the two front rooms. They had knocked a bag on the door of the other front room and it was swinging. I stuffed my feet into my shoes and ran round to my Nan's house in tears. Everybody thought that I was being silly.
My Nan swears this is true. When her mother died (I was 7) things happened in the house, especially in the leading up to her funeral. Mirrors just dropped from the walls, footsteps were heard, gnomes in the garden were moved, etc and there was a lot of drama over getting something big out of the house which would require the front doors to be opened, and they'd never been opened by my great grandmother so the strange things intensified. I think it may have been her coffin or bed or just something significant to my great grandmother.
My Nan moved into the home after the funeral and the weird things continued, along with boots being heard on the stairs, which apparently my Nan's father wore around the home.
A few years on, the room that had been Nan's mother's bedroom downstairs was a playroom, toys and games everywhere. There were no photos of the room as a bedroom anywhere, I don't think that any photos of my great grandmother were up either. My young cousin, born soon after GG died, was in the playroom alone. She came out and supposedly told my Nan and her mother that "Nanny P" was in there. P was my nickname for GG. She went on to explain what the bedroom looked like, in detail, and say some things that she wouldn't have overheard. Sends shivers down my spine to this day.
My Nan again, must be about four or so years ago now but her fostered/adopted daughter woke up one night completely inconsolable, saying something about the TV in her room. Nan calmed her down and put her back to bed. Not long afterwards the same thing happened and she refused to go back into the room. In the next day or so my Nan had a phone call to say that ADD's mother had killed herself. ADD took months to even go near the room and even then, would only go in during the day the last time I asked about it.
I believe the big black cat stories, there are a few big wild cats around. The Beast of Essex was a common story when I grew up and a few kids spotted it in Colchester a couple of years ago. I hear that there's a lion in Essex too but that may just be a big house cat 