Check with your dealer or manufacturer to see if synthetic is usable in your car first of all. For example, RX-7s have traditionally recommended against synthetic because they inject some oil into the combustion chamber and synthetic oils don't work as well for sealing the "apex seals" in this application.
Beyond that, there is also a theory that synthetic oils may break down sludge and varnish which is sealing what would otherwise be leaks. So switching to a synthetic may cause a car that has no leaks to start leaking. I've never had that happen or known that to happen, but they do promote reducing sludge and varnish as a feature of synthetics.
However, if you have a turbocharged car, I'd use synthetic in it in any case. Turbochargers run at extremely high temperatures and are lubricated and cooled by oil. Synthetics react better to extremely high temperatures, so they are particularly useful in turbo applications.
That said, I've had some mixed results with Synthetics that make me question their worth.
One situation was an old '86 Subaru GL-10 wagon I had. It had a turbo and I put Mobile-1 in the engine and Red Line in the trans-axle. My wife and I took the car up to Canada on one trip where it was -25 to -45 degrees. The car ran surprisingly well, including shifting, compared to my experience in Colorado which rarely gets below -10. That was a clear win for the synthetics.
A decade ago I had a '89 VW Vanagon that I put a new engine in. After the initial break-in, I ran nothing by Mobile-1 in it. I had all sorts of problems with it keeping oil pressure up, which went away when I switched to a high quality regular oil (Delo 400) at a mechanics' recommendation.
Around 5 years ago my '97 Audi, which I had run from the first oil change with Mobile-1, had 2 of the lifters go quite badly, in what I'd consider a lubrication-related failure. Here is a picture:
Now, maybe this was a design problem, but I haven't heard of it being common on these engines. SO my take on it is that, in my admittedly small sample size, the only lubrication-related failure I've ever had has been on the car I ran Mobil-1 in from the first oil change.
So, I won't run Mobil-1 in my cars, and I'm mostly skeptical of it's competitors.
I'm now running the Delo 400 traditional oil in my Audi that had the lifter problems -- around 100K miles since the lifter replacement and no problems. In my other car, it takes some fairly unusual weight, and recommends synthetic, so I'm running Red Line in there, at around $10/qt. Lucky for me it "only" takes 8 quarts for an oil change :-). But, it's an expensive engine, with 40 valves, so I think it's worth the "good stuff".
In my transaxles, I've been extremely happy with Red Line, so I'm trying it in this engine.