TL;DR. The short answer is, it depends. How old is your bike? What was your bike designed for? How was it maintained? What are your driving habits?
It depends on a lot of different factors. The main factors being; age (of the bike), history (What were the various owners like? Tracked? Daily driver? Commuter? Stop n Go or Freeway? etc.) and the final main factor being design of the motor.
If your bike is more 15 years old and you don't really know its history (i.e. how/whether it was well maintained) I wouldn't necessarily suggest switching to synthetic oil. The reason being, that (high/premium quality) synthetic oils often have very small and uniformly sized molecules and much lower viscosity (on average) than conventional oils of the same weight. That can sometimes result in seepage or even full on oil leakage on an old motor with dried up, hardened old oil seals. Especially on a low mileage, older bike that's been stored for several years. Not always, but I have seen many owners complain that after switching to a synthetic oil, their previously "perfect" bike now leaks oil.
Now to be fair, many owners don't realize that when they "switch" to a synthetic oil they often use a different weight oil than they did previously. The most common one being Rotella T6 5W-40. Which is a great moto oil. You just have to realize that it's a "thinner", less viscous oil than is usually spec'd for most bikes. Most are usually spec'd for 10W-40 unless you live in really cold or really hot climates. Most bikes do just fine on a slightly "thinner" or even a slightly "thicker" (20W-50) oil. I'm just saying that you need to experiment and pay attention.
Also, a LOT depends on how you drive your bike. Pop wheelies a lot? Slam through the gears while pretending to be Valentino Rossi on Redwood Road every chance you get? Change your oil more often. And check your various bolts, nuts and fasteners.
Driving gently and doing lots of short rides/errands where you're constantly starting and shutting off the bike creates other issues. You need to check the battery more often because it likely never gets a chance to recharge properly. It also probably never gets hot enough to really burn off excess water (from condensation) and probably doesn't get to full operating temp which can lead to excessive sludge buildup in the engine because the oil is still just warm, then cools off and "pools" inside oil galleys and passageways.
Like I said, it depends.
What I mean by design of the motor is that there is a huge variation on internal tolerances and size of oil passageways depending on what that particular manufacturer designed the motor for. Race-bred engines (think Ducatis) usually require higher quality, premium oils because they have smaller/tighter internal tolerances. Bikes that are purpose built for commuting or touring generally speaking aren't as demanding. There are exceptions everywhere, but for the most part, this holds true.
Bring some to top up when needed.