I've always just assumed he did, but I wonder why? Obviously Abigail thought so, but we don't really know what it's based on.

She mentions how upset she was that the police didn't investigate more than they did. Knowing her personality, this means that she definitely would have made her suspicions known about Edmund, including any evidence she had. Plus, being a celebrity she would have had some influence. But whatever case she had, they rejected it. So it's possible Edmund didn't do it.

It also makes me wonder, any suspicions Abigail shared with the police, it would have likely got back to him. Right?

Yes, I believe it was supposed to be apparent that he had been guilty. The part where they showed him pick up her picture and kind of smirk a bit was the indicator. They didn't show him looking at her with sadness or like he was heartbroken. And it hadn't been that long since she had died. And his motive, possibly even for marrying her in the first place, was the rights to the play that she owned which presumably was very valuable. Also, although Abigail probably told the police what she thought, it would have been a tough case to prove (the body wasn't found), and Abigail knew at this point he would never get charged, let alone convicted.

Great points. I never picked up on that. Even if the author didn't voice concerns to the police - which I am sure she did - you would think that Edmund would at least suspect that the great mystery author knew that he killed he niece. Or at least suspected that he did. Now that you have brought this to light I have to wonder why Edmund would be so trusting of Abigail. Also that he would have to feel uncomfortable around her.
Besides the smirk, which tells much, the last few seconds in the safe tells a lot. Abigail asks, "You think I didn't know." He does not scream out NO or KNOW WHAT? or I DIDN'T DO IT. Seems his reaction is a bit of 'you got me' and surprise.

(Personally if I thought Abigail even suspected that I killed her niece and if I HAD NOT I might have had a sit down with her and got everything out in the open. Obviously Edmund never did.)

Of course he might never have wondered about Abigail's suspicion if he actually did not commit it.

One telling point, at least to me, is that when Columbo mentions the accident (during the scene with Abigail and Dog on the boardwalk) he notes that it must have been painful for Abigail. It is almost as if he already knows her motive for the murder of Edmund. He does not say if he thinks Edmund is guilty or not. I think he would have said something if he thought Edmund was innocent - to let Abigail know that HER crime could have been avoided. Just my opinion.

In a way, that smirk is my problem with suspecting him, because it's about as close as the story comes to anything concrete. I still don't think Edmund's apartment not having a single photo of Phyllis really counts all that well, since maybe genuine grief had caused him to get rid of them, or put them away who-knows-where. In some other place, I mean, not the apartment.

And like Clay, I've always wondered how Abigail's suspicions didn't get back to Edmund. Even one of her lines TO him is meant to sound like an accusation - when he reminds her how he searched for Phyllis, her line is "I know what you did. Everything you did."

I believe that Edmund did INDEED kill Phyllis.It just seems to me that if my wife had drowned only 4 months before, I would begin to show emotions everytime someone mentioned it. When Edmund walks with Abigail near the water,he seems to show no emotion. Also, the way he looks so indifferently at Phyliss' picture, in my opinion, bespeaks a cold blooded individual!

That is obvious that he killed her.