I would examine the seemingly most unlikely coincidences, attempt to ascertain all relevant facts regarding each sequence of events, and then try and estimate what the odds against that sequence were.
If you can prove that any one of the the events was impossible then you're on to something, otherwise what does this show? Even assuming you can work out the probability of the events all you have proven is that a sequence of events that was possible did occur. There is no predictive application as you are dealing with single occurances.
That aside how do you propose to calculate the odds? This is where the 'Texas Sharpshooter' fallicy that Joe has repeatedly refered to comes in, as there is no definition of what constitutes a 'syncronistic' event before time you have no way of calculating the range of possible events that would have constituted a 'hit' had they occurred but didn't.
For example, using Quarky's illustration "You're talking to your sweetheart about something fairly obscure...say, an old song. You turn on the radio and its playing." Simply because it's something we've all experienced at one time or another and it's a fairly simple one to discuss as an academic exercise.
How would you calculate the odds? I presume you'd take the odds that these two people would be talking, then that they would talk about that song, and then that they would switch on the radio, and finally some combination of song duraton against airtime, number of channels, frequency of the song being played etc? I'm not after figures here, but is this your general line of thought?
Looked at this way the odds are astronomically against this happening, but calculating it like that would only be correct if the event was predicted ahead of time, this is only correct ahead of time with a specific outcome, those two people, that song, that time etc.
A more accurate view of the situation is that people talk all the time, sometimes they will mention a song, sometimes they will turn on the radio each of these events happens millions of times every day, sometimes they coincide, sometimes two coincide. The window of opportunity is further widened, if someone turns on the radio and the second song that plays is one they were talking about is that syncronicity? Third, forth, the radios been on all day but they're sure they hadn't heard it on that channel for absolutely ages? They didn't speak about that song, but it was the theme to a movie they watched last night? The title's the same as a different song they were discussing? The song seems appropriate to something completely independant, but they can see the connection?
Billions of people, billions of possible connections but all in the eye of the beholder. And is one possible coincidence out of the infinite number of things that could appear significant after the fact.