Stopping Problem

Your filly sounds just like my little filly that I am working with now. Here are a few things you can do.
 
Work on side to side flex. Getting her to flex with the bit from side to side. (On the ground then in saddle) If she decides to walk off on you or not stop, this will show her you still have her head and you make her stop. After some time with this, she will have no way of doing any thing or else she will just keep going around and around. Once she gives and stops moving (if she does move) reward her then walk, then do it again.
 
After a bit of this. Use the fence or a soild thing in the arena for something that she is going to have to stop. Walk up to the fence. Keep her nose in to the fence. Ask for a stop. Once she stops reward her. Then do it again.
 
 

You say she's only been broke for ten months and standing several of those?  Do you remember when you were in grade school and came back from summer break?  Didn't remember much at first did you?  At least I didn't. 
 
In this case, I'd say you are asking too much of a green horse.  Put her back to the basics as if she isn't broke at all.  You really need to start over again.  Admittedly, it won't take as much time to go over the same ground, still you need to.  The suggestions FreeMare made are excellent and would be a great addition to the refresher course.  Still, I think you're pushing a little too hard and it's somewhat unfair to give the filly so much time off then ask so much of her.  Remember, she's still a child.  Approach her that way and be patient and you won't be disapointed.  Give her another year or two before you expect her to be a reliable trail horse.  Patience is a virtue, that with horses, is often richly rewarded. 

Great advice so far.  Remember I have two vids up on building a better stop and they go over the basics.  From the sounds of it she is setting up on the bit, refusing to get soft at the poll and in the neck.  There are many reasons that she could be doing this such as nerves, bit doesn't fit her mouth right, teeth...that sort of thing but first and foremost you need to get her soft again and really giving to that bit.  Work on lateral flexion and vertical flexion at a stand still to help her better understand what is being asked of her.  Sometimes the pop in the front is a bluff to get you to release the bit pressure without her giving to it.  Make sense?  Really work on your one rein stops and bending to a stop till she starts paying attention to your body though for at least a week before asking for her to stop straight on.  The stop should not be a forced thing, they should want to crave that stop and be soft doing it but it takes consistancy and lots of time to get it good.
 
Good luck and keep us posted.

@FreeMare    I do side to side flexes every time I work her, but on the ground. I haven't done them in the saddle like I normally do, mostly because she keeps going around and around and then gets annoyed with me. I will try them from the saddle now, and then try what you suggested with the fence.

@studclunker    I did exactly as you said when I first started working her, and she really surprised me with how much she remembered. Of course, I went slow and treated her exactly as you said; as if she wasn't broke. We started with just the saddle and some working in the pen, then moved on to lounging on the line before I did some flexing and got back in the saddle. I do a lot of desensitizing with her as well, because she can still be a bit spooky. But as you say, she is a child. I'll stand there and throw my lead rope over her until she stops jumping all day. We went for one three or four hour trail ride, which she was very well behaved on. She loves going on them. She does everything else I ask; this is the only trouble I seem to be having with her.

@trainer101    I'll be sure to watch those videos. I understand the popping better, and will try to not be as intimidated by them. I'll do some of the one rein stops. I try to teach the whoa as something that they want. It's time they get to stand there, do nothing, and catch their breath after all! My gelding absolutely loves to whoa for me. He's an awful lazy boy, and I'm not used to the spirited little thing that she is!

Mostly, I've been working on the stops in the round pen and getting her to slow down in general. She's very 'hurryhurryhurry', and I want her to sit back and relax with me and not be so rushed. When I did get her to stop, she started pawing and then backing up quickly. But next time I work her, I'll definitely try what you have all suggested and try to put a nice stop on her.

Thought you might want to know how she was doing. She didn't get worked for a little while because my grandmother passed away, but now we are back on schedule.

I worked her for a few days without bothering to get on her, and for good reason. She decided to buck the first time I put the saddle back on her. However, with a little work she did fine with the saddle again and we had no problems. I did a lot of flexing with her from the ground during this, working on getting her to stop better at whoas from the ground as well. After a while, I went ahead and got back on her.

I started working on stops. She had a bit of a fit when I stopped her, and she tried to go forward and so I bent her head toward my knee to make her stop again. However, after that, she stopped much better for me. I did a lot of flexing in the saddle, and some of what Freemare suggested with stopping before the fence. Overall, she was doing much better.

So we started trail riding again. I've taken her out on two rides, one was just walking, one was just walking and trotting.  We went through a few puddles the second time and she did very well over all. Both times I had her lead for a little while, and we stopped as we went down the trail and waited for a little while. I told her she was a good girl, and off we went again. I'm planning on doing more with her on the trail tomorrow.

That's wonderful that you are feeling success with your mare!  Hope she keeps up the improvement

Glad to hear things are going well with the mare.  Sorry to hear about your Grandmother.

Sorry to hear about your grandmother.  I lost mine 2 years ago.
Just curious, the bit you changed to?  Would it be a Tom Thumb?

Glad to hear that my info helped you out. Sorry to hear about your grandmother.