I get ideas for the beginnings of stories often, mostly the beginning of November (thanks, Nanowrimo). I then write a fabulous page or two, and lose the story. Positively awful. My problem is the plot, which I suppose is a pretty normal problem.
I really like some of my beginnings though. I had one where I was writing from the perspective of a toddler. I felt like I was capturing it pretty well (ha, if one can capture the thoughts of a toddler...), but then after a page and a half, I decided it wasn't going anywhere.
So I instantly started a new story about a dragon. That was even shorter (only three paragraphs).
Then there is the story I titled "Until the Last Rose Fades", written in first person, and I got four pages into it. That's basically a record for me. But four pages in I had a love interest, but no plot, and no skill for writing romance.
Now I am writing "Ada". I'm about a paragraph short of the length that "Until the last Rose Fades" died at, and I'm still going strong. I've got the whole plot lined out, and most of the story. I'm a bit shady in some of it (lots of it, rather), pretty much any time that the main character stays in one place for a while, but I do know what I want her to accomplish in that time, so I think it will be okay. I'm pretty excited about it. It is nice to be working on a story that I think may actually work.
I'm considering posting it on my blog as I progress, but that might be over-sharing on my part. It isn't very good (ugh), and I wouldn't want to subject you to it. However, I'll probably tell you occasionally how it's going, like when I reach page 10, etc. It's kind of hard imagining me actually getting to page 10 on a Word document....