Few topics in motorcycling provoke as much debate as oil. For what they're worth, here are my opinions. Actually, they're more than just opinions, since, as you'll see, I back them up with research from various sources. You will find plenty of articles on the web written by private individuals with various degrees of expertise (anything from the guy next door to professional petrochemical engineers). I am just a guy next door, not a professional petrochemical engineer, but I do try to provide reliable sources for my opinions. Check to see what others are using to support their arguments.
My main opinion about oil is this: there once was a time when a decent car oil was good enough for motorcycles in most cases , but that time has passed. Nowadays, you should use motorcycle-specific oil. Typical car oils are no longer appropriate for motorcycles.
The most important reason that you should use motorcycle-specific oil rather than car oil has to do with the fact that the typical motorcycle with a wet-clutch design uses its engine oil to lubricate the gearbox and the starter mechanism. I know what you're thinking -- yeah, yeah, the same old stories about transmission gears chewing up the oil -- but there is more to it than that. Those arguments, and many more like them, applied equally well years ago when it could be argued that car oils were good enough. But circumstances are different now, for specific reasons.
The overall reason is that the formulation of modern car oil is optimized for increased fuel economy, which is achieved by the use of friction modifiers that make the oil more slippery. Such oil is so slippery, in fact, that it can cause problems in motorcycle engines, and that in a nutshell is why you should not use it. Now it's time for details.
Automotive oils are classified by the American Petroleum Institute (API). The current rating is "SL" (at the time this article was originally written). On a typical bottle of oil you'll see the API emblem, with the letters SL inside a circle, and the words "Energy Conserving II" around the circle. Do not use this type of oil in a typical motorcycle.
A research study  published by the Society of Automotive Engineers (SAE) and written by researchers belonging to the four Japanese motorcycle makers looked at then-new SH-rated car oils to evaluate their suitability for motorcycles. Among their conclusions:
A column in one of Honda's technical service bulletins  gives an update on SH oils. I reproduce it here without permission.
This article appeared in 1998 and would apply to oil classifications that follow SH including the current SL rating.
"This type of oil was developed to optimize the kilometres per litre attained by automobile engines. To achieve this, most SH oils contain friction modifiers that significantly reduce frictional losses on internal combustion engine components. [...] While SH oil is fine in automobile engines, the situation is different for motorcycle engines.
"The use of SG  rated oil with friction modifiers in Honda motorcycle engines may cause the following problems:
- Wear between the camshaft lobes and rockers arms due to the lubricant breaking down between the two components.
- Slipping and deterioration of wet multi-plate clutch systems.
- Slipping of one-way starter clutch systems.
- Possible wear and pitting of transmission gear teeth due to the decreased shear-stability of the oil
"Not all SH rated oils use friction modifiers. But since oil producers are not required to state if their oil contains friction modifiers, it is difficult to tell which brand may cause a problem. With this in mind, we recommend that SH rated oils not be used in Honda motorcycles."
So, if you should not use SH (or later SJ, SL) car oil, what should you use? Presumably a motorcycle-specific oil. But when you look at a bottle of motorcycle-specific oil you see that it has no API emblem on the front. Does that mean it is not a respectable oil? No. I learned from a spokesperson at Spectro Oils that the API does not allow the use of the API emblem on the front of the bottle if the oil does not meet the current standard. So any proper motorcycle oil will not have an API rating on the front. If you look on the back, you'll see that the oil is identified as SF or SG or whatever. There is actually one exception to this, and it's particularly good news.
The good news is that in recognition of the fact that car oil formulations are no longer suitable for motorcycles, there is a new motorcycle-specific rating designation: JASO MA. Some of the leading oil makers, such as Mobil and Motul, are offering oil with this designation, and the corresponding new API emblem appears on the bottle.
None of the arguments I've provided so far has relied on the familiar old ideas about motorcycle engines being harder on oil than car engines that are so contentious in the debate. What I've presented so far is sufficient to show why you should not use modern typical car oil in typical motorcycles. But let's at least mention some of the old reasons.
Zinc is an anti-wear additive typically found in motorcycle-specific oils at higher levels than in car oils. The chemical that provides it is ZDDP, zinc dialkyl dithiophosphate.
This ingredient is helpful for preventing gear wear. It is used in motorcycle engine oils to help protect the transmission. But phosphorus is bad for catalytic converters, so car oils, which are not used to lubricate transmission gears, have very little phosphorus.
The common car oils are 10W30 (or 5W30 and 0W30 which are more pourable at cold temperature). These 30 weight oils are too light for motorcycles. Most motorcycles call for 10W40 or heavier in order to support bearings, resist against consumption due to volatility, and so on.
When the motorcycle manufacturers themselves, who have no vested interest in whose oil you buy, warn against the use of car oil for specific reasons, and when the oil industry itself acknowledges this difference in requirements by responding with a new set of standards for motorcycle-specific oils, I think it is pretty safe to put your trust in the recommendation that you should not use the modern SL "Energy Conserving" oils in your motorcycle.