Here is the closed rail one.

I actually prefer the closed rail for vinyl fencing. It looks like it is what it is supposed to be, rather than looking like a vinyl fence pretending to be a white painted wood picket fence. But I wouldn't be surprised to be in the minority opinion on this one. ;-)

Here is another closed

And another open. Sorry if too many pics just trying to paint a good picture.

Closed rail looks more current to me. I would pick it.

I prefer the closed. Unless you have an uber-traditional cottage type house. Then I'd go with the other.

I personally prefer the first picture you posted. But, I think it totally depends on your style of house and neighborhood. Either style has the potential to look out of place. Are you in a neighborhood setting like most of the pictures you posted (except that one that is obviously rural)? Are the houses in those pictures similar to your style of house (mid-range typical neighborhood and not million dollar brick mansions)?

Closed rail fence. Simple strong clean lines.

I prefer the open. IMHO the closed one looks like a railing.

Here's a picture of house fence will be placed on back yard.
Thanks for opinions. Please keep them coming.

The first closed fence. It is more similar to the one in the front of the house.

I completely agree with Gyr_Falcon.

Also, the closed fence goes along with the railing at your front door.

Closed rails would be my preference, but have you considered the aluminum fencing in black that resembles wrought iron?

Thanks for posting a pic of the house. I vote closed in that setting.

Closed. It also looks a bit more elegant than the picket plus it goes with the railing on your porch.

What Graywings said! Check out Echelon, you can get 8' lengths that are designed to give a little for grading differences. It is also very easy for a DIY and excellent quality . I have personally had that fencing and highly recommend. Large expanses of white fencing only look good on a horse farm estate.

Agree with Graywings too and was going to suggest black wrought iron. Her suggestion looks like a good alternative to the wrought iron. Black looks so much better because it visually disappears. Is this an option in your neighborhood?

Are there other fences close by? If so, I would probably use that pattern. However, I like the idea of the black. White is pretty when it is clean. But the black will be more forgiving in looks when it starts getting faded with age.

Thanks All. Yes our HOA limits us to white vinyl so not much to choose from. Went closed. Will try to post a completed pic.
Thanks Again.