Marriage is a learning experience, as is life in general. People who are married or have been married see life a bit differently than those who have never married. And I'm sorry, but close does not count in this regard. The fact that you have lived with someone for an extended period of time, or were once engaged, is just not the same.
A person who is widowed has a different outlook on life than someone who is divorced. Both of them have different outlooks on life than someone who has always been single. What the divorced and widowed person have in common is that they were once married, and lost their spouse in some manner.
Also, there are different types of divorced people. Some have only been divorce once, others have had several spouses over the years. And the number of years is significant. A person who has had three spouses over a period of 25 years is different from one who has had three spouses in three years. (I might also be more than a little bit concerned about someone who has been widowed several times in a few years.)
Time brings another aspect to this discussion. How long a person has been single, divorced, or widowed is a significant factor. Someone who has been divorced or widowed for many years will be more like a person who has always been single that like a person who is recently divorced or widowed. I don't know a lot of widowed people, so I can't say a whole lot about them, however, in the case of divorced people, someone who has been divorced a couple of months is quite different from one who has been divorced for one or two years. And that person is different still from one who has been divorced ten or fifteen years, maybe even only five years.
What I find interesting about the recently divorced is that they are easy to pick out in a group of singles. I am going to use an ugly word to describe them, desperate, but I challenge you to come up with a better one. It may be that these people need a little more time with a support group before they can become successful singles. But the reason I suggest that these people stand out in a singles environment is that they act in a way that is almost annoying. They lack the finesse that others have, they are not yet comfortable with their new single status, and they are determined to pair off with someone right now. And yes, I was once one of them, so I speak a bit from experience here.
I have already made the point that these recently divorced people may need a little more time with a support group, rather than a singles group, but there is another more universal point. The ideal person for someone who is recently divorced is another person who is recently divorced. They share the same outlook on life and they may actually be quite good for each other. I should also caution you that there are some people who prey on these types. Some people recognize that those who are desperate are also "easy."
The universal point I am trying to make here is that like individuals are more compatible than disparate individuals. A widow may be more compatible with a widower than a single, in the same way that two singles may be quite compatible. Time affects this, and so someone who has been widowed for ten years may be quite compatible with someone who has been divorced for a similar length of time, or with a single. As time goes on, you are less likely to be compared to the ex- or late spouse.
At this point a disclaimer is probably in order. This discussion is based on observation and personal experience, however, nothing in life is absolute, not even life itself. So do not necessarily rule someone out because of a difference in "singleness." But do be aware that this difference may be a factor.
Be honest about your own situation, and be open-minded about theirs. If you are reading a singles ad, realize that SWM, DWM, and WWM may be three very different situations. But remember that time is a factor and that in some cases these may be more similar than they first appear. If you are placing an ad, and if you have been single for a while, then you might note that you are seeking (or in search of, which is ISO) a D/SWF if you are a DWM (and if race is not important, leave out the W, B, H, or A). But beware of ads with DS in them. Determine the context, because in one case they may be seeking a Divorced or Single person, in the other case they may be Desperately Seeking a warm body. If someone is widowed or divorced, be sure to ask them how long that has been the case. It may or may not be significant.