I have had a few experiences. I'm not sure how I feel about them...I certainly don't embrace it, but I don't discount everything either.
When I was a child (between ages 2 and 4), my mother would come to my room to find me staring out the window talking to someone. When she would ask, I would tell her that I talking to my aunt/godmother who had recently passed away. This went on for years, though I don't remember it myself.
When I was about 8, I remember sitting in my bed reading one night, when I just had this urge to look out the window. When I looked in the window, I saw a woman's face. It was comforting and familiar, but I couldn't place it. When I described it to my mother next morning, she said "That's Donna!" and told me about my experiences staring out the window talking to her as a child. I haven't seen her since. She died shortly after giving birth to her son...I think she stuck around to look after him and just popped in on her god-daughter from time to time.
My parents live in an older home, and my mom has had a few experiences there. There is an older gentleman who still resides there apparently, and he doesn't like for new people to stay there I guess. When my parents moved in, odd things happened and she would see him, but then as he got to know them, it settled down. They moved in there when I lived in another state, and the first time I went to visit them and stayed in the house, I sat down in the bathroom to get ready to do my makeup, when an air freshener went flying across the bathroom at me. It missed, but it freaked me out. I told my mom and she just laughed and told me about everything she had already seen/felt there and said not to worry about it. I still don't like going into certain rooms there, haha.
I've had a few other things happen, seen black forms pass by or out of the corner of my eyes...but I usually second guess myself.
I don't know what to do about this one or how to react to it. My youngest daughter, about a year ago, was playing in our play room. She suddenly starting hysterically crying so I ran to the room and she was just screaming and shaking to the point where she couldn't move. When I finally got her to tell me what was wrong she insisted she saw a "glowing green man" come through the wall. A year later she still talks about him and tells me she sees him sometimes if she wakes up at night. I want to say it's just nightmares, but to have started during the day while playing alone one day? She's only 3. And who am I to say she's not really seeing something just because I'm creeped out by it?
My oldest daughter will tell me when someone is about to call, or something that has happened or is about to happen that she would have no way of knowing. A couple years ago, my husband was deployed to Afghanistan. His cousin was in a car accident and passed away. I didn't share this information with my daughter, she was only a year old (almost 2). One day she told me "Mama, daddy crying. Daddy sad." I got a phone call from my husband a few hours later letting me know he had received the news...a few hours earlier, and only then had a chance to call. He had definitely been crying, I could hear it in his voice.