The manufacturers stock recommendation based on regular service intervals and "normal" driving is as follows:
Temp Range -20 degC to 0+
10w-40 Semi or Fully Synthetic
10w-50 Fully Synthetic
15w-50 Fully Synthetic
10w-60 No recommendation
There are always options to consider with regards to "special requirements" if you have modded or re-mapped your car or are using it for trackdays and spirited driving.
Specs have moved on now and most oils sold today are API SJ/SL which is not a problem, in fact the API tests are more rigorous nowadays.
As mentioned above, there are reasons for using different oils but you should never stray too far from the recommended grades as they are the most suitable for your car. For example 0w would not be good here nor would 60, it's just a step too far.
So, what are the reasons for choosing a different grade?
There are advantages to going thicker or thinner so long as you don't overstep the mark.
5w-40 instead of 10w-40
5w offers better cold start and short journey protection as it's thinner than 10w and remember this is when 70+ percent of the engine wear occurs.
This is a good choice if you are doing a lot of short journeys and cold starts and don't do regular track days or competitive driving. (unless you choose an ester which I'll explain later)
10w-50 instead of 10w-40 - 15w-50 instead of 10w-40
50 offers better protection than 40 when the engine is running hot so if you're doing track days or hard driving you may want to consider this option. The best one for your cars would be a good "shear stable" 10w-50 which would give good cold start 10w and sae 50 protection when running hot.
A word of warning about 60 grade
Modern engines do not need a 60 grade, it's really too thick and, even hot-running engines do not need SAE 60 oil these days. (by ‘hot’ I mean 120-130C). SAE 60 is heavier than most SAE 90 gear oils. If an oil is too thick, it de-aerates slowly, leading to cavitation in the oil pump, or the bearings being fed slugs of air along with the oil. Also, in normal driving it's harder for the engine to circulate and causes poorer fuel economy and power deliver. Thinner oils deliver more BHP. (oil drag)
Synthetic Oils instead of Mineral Oils.
It is a fact that Semi and Fully synthetic oils are better for your car, they flow better when cold and give a greater level of protection when the engine is hot.
It is a fact, not myth that complex engines such as DOHC, Turbocharged, and Supercharged when driven hard will always have a greater degree of protection from a synthetic oil.
A synthetic oil is more robust and will last longer (that's why they cost more) mineral oils need much more frequent oil changes but they are cheaper. Semi-synthetic is to some extent the "middle ground" for pennies more than a good mineral oil.
Ester Synthetic Oils against Normal Synthetic Oils
The oils that give true benefits are ester synthetics (pretty hard to come by as they are expensive to make). The reasons for their application in motorsport and the aviation sector are clear:
They assist the additive pack in a motor oil formulation because they are surface-active (electrostatically attracted to metal surfaces), so they help to reduce wear and friction.
They are fluid at very low temperatures and at high temperatures they are very chemically stable and have low volatility (don’t evaporate away).
They also help to prevent hardening and cracking of oil seals at high temperatures.
It should be said that there are three types of synthetic oils on the market and they vary in quality and price the lowest being MC/HC (Molecularly Converted/Hydrocracked) mineral oil, POA's Poly Alpha Olefins and Esters.
So, there you have it, some advantages and disadvantages to consider.
What would be my recommendation?
Well, putting brands aside as this is personal choice and they're not always as good as the marketing "blurb" will have you believe I would say that applying the following rules would be the best advice.
Low Cost Option
Good quality 10w-40 Semi Synthetic which has a good VI Index and is shear stable
Middle Cost Option
Good quality 10w-40 Fully Synthetic which will be shear stable
High Cost Option (but belt and braces protection)
PAO 15w-50 Fully Synthetic
Ester 15w-50 Fully Synthetic
Ester 10w-50 Fully Synthetic
At the end of a day it's a minefield out there and generally, you will always get what you pay for as good oils are expensive to make.