Just like anything else.... it depends.
IMHO, 3 shot groups are like rose colored glasses. If you shoot enough of them, you will see what you want to see - ie an accurate rifle/load. If one shot opens up the group, it is called a "flyer" and more 3 shot groups are shot until you get a picture worthy of bragging about on the internet.
Multiple five shot groups give you the ability to get an aggregate average over time. Obviously if you're using a chronograph to figure out SD and ES, that magic number for a statistically valid results is around 30. It doesn't take too many trips to the range to get six reasonable groups of five to be able to have some reliable data. You can see your data come to life, like a group that opened up to X moa because of the 15mph wind, and so forth.
I do like ten shot groups because very little can be hidden. Almost
nobody likes to post their 10 shot group totals probably because most people don't have a rifle / ammo / skill combo that can shoot ten sub-moa (and who is going to admit to the internet that they aren't a sub-moa shooter, right?). I'm not saying that I have this ability, but who likes posting 10 shot 2-2.5moa groups? The larger issue with 10 shot groups is the shooter's ability to make consistent shots. Concentration and your routine become more important and the question becomes - was it the rifle and ammo or the shooter?
So, for heavy recoiling firearms, I still like 5 shot groups. You can aggregate them and it gives a shooter time to rest and barrels time to cool.
Light recoiling firearms, especially for load workups, benefit from 10 shot groups to see a worst case scenario. Multiple 5's can get you reliable data.
For rimfire and other super-light recoiling firearms, 10 shot groups really tell you how good your ammo is and many fliers are in each box.