I definitely believe this, and have a few of my own to share.
My Mum steadfastly believed in the afterlife, and she said that when she died she would make a picture fall off the wall. She died in her own home, with me, my sister and our Dad there. My sister went home that night and a canvas that had been hanging in the kitchen was on the floor.
Recently, I was watching a video made by a psychic and he said that you should ask your loved ones for a sign, and tell them what the sign is. I secretly asked my Mum to show me a butterfly. On Mother's day, my DH always gets me a card from the dogs, and it was a plain card saying "Happy Mother's Day" with one butterfly pin badge in the centre of it. I told my sister this, and she told her 10 year old daughter, and her daughter said "If I was thinking of Nanna, I'd think of the moon". Later that day they went out for Mother's day dinner, and the lady who greeted them at the restaurant had a t-shirt with a moon on it, moon shaped earrings and a bracelet with moons on it. Suitably freaked out (!) when they sat down, my sister asked her daughter what she would think of for the other Nanna (also passed), and she said the sun. The waitress appeared and had a huge tattoo of the sun on her inner arm!
A few months ago, I decided to see a psychic with friends. I've never done this before. A few days beforehand, I kept asking my DH if he could smell the strange odour on our house - it smelled like wood. Couldn't locate it. Anyway, when I saw the psychic, she told me my Mum's name and told me that my Mum had placed a smell in my home - not perfumey, but woody! She also asked me if I had sent a saucy pic to my DH - a weird question to ask a middle aged woman I thought - and although I denied it, this was actually the truth. She also told me that someone who had recently been talking about adoption would not go ahead - this is true - my son and his girlfriend had been considering this, but they had just split up. Lots more too.
My Mum went to a psychic fayre around the time I was considering leaving my first marriage. I had not told a living soul I was thinking about this. A stall holder beckoned my Mum over and said "I have a message for X" with X being my name. I have met around 2 other people in my life with the same name as me, that's how unusual it is. Psychic went on to tell my Mum that I would be moving house and finding a new man......Mum thought no way, she's happily married.....of course it was ALL true. That psychic also told my Mum that she would come to own a guitar under strange circumstances, and that she would win the lottery. Some time later, my parents were sitting in the garden at midnight, after having had a BBQ, and a guitar came flying over the back fence, and landed squarely on the lawn. Shortly after that, they won £20k on the lottery.
Oh, and one day, me a DH were moaning a bit about his Mum (passed), and a canvas on our stairs flew off the wall!
I cannot fathom any logical explanation for any of it, but when I've told these stories on here before, people are tripping over themselves to disprove anything freaky has happened, but how on earth can you explain any of it?