Building settlers at size 3
Wait- this is a bad idea?
Sometimes it is and sometimes it is not. Sometimes it is better if a city is larger when it goes through cycles of growth and shrinking while building settlers.
Sometimes you just cannot afford to wait, and have to get out a settler as soon as possible. In other cases you cannot afford larger cities, for example in corrupt lands when you have only one lux and one citizen born content and are still in despotism.
I would say, in the early phase it is always a bad idea. (The exception that Lord Emsworth mentioned, i.e. corrupt towns that can't afford to grow, happens usually only later, when you already expanded a bit.)
The reasons are:
1. Building settlers can be done much faster at size 5-7 than at size 1-3, because of a bigger town usually has more food and more shields per turn. People think "I must expand as fast as possible, so I need to get that settler out as fast as possible" and they build it at size 3. But in fact, settlers can be build much faster and consequently the expansion will be much faster, if you are patient and build the settler upon reaching size 6 or 7.
Example: a town with "limited" food resources (i.e. 3fpt tiles). Just consider the case where you have two wine resources or two floodplains at your disposal. The small town then needs 12 turns to grow from 1 to 3, while the big town can grow from 5 to 7 in 10 turns. So here we already see a much slower settler rate!
2. What scratchthepitch said.
I found that research suffered because the town only averaged about size 2 when building settlers timed to finish at a size 3 town. Research was sped up by timing the settler to finish when the town reached 4 or 5. But sometimes it's better to expand sooner, in which case you cant wait those extra turns for the towns to reach that initial 4-5 pop to finish the settler.
Here I might add: usually that "waiting until the towns reach that initial 4-5 pop" has to be done only once! So settler 1 is built lets say 6 turns later, but all following settlers 2 - n are built at least at the same speed, usually much faster. And even if the first settler is built later, it does not mean that the town is built later as well! Usually in the early stage the road network is still quite "non-existing". So what I often see is: people rush out the settler at size 3, and then that poor guy spends the next 6 turns travelling to its destination at 1 tile per turn. If the settler would have been built 6 turns later, the workers could in the meantime complete 2-3 missing road segments here or there, and then the settler can reach the same destination in only 2 turns instead of 6. So even if the settler is built much later, often the town can be founded around the same time (or only insignificantly later)!
To illustrate the point that scratchthepitch has made, let me give the following simplified example: assume our town has 2fpt at all sizes (so no food boni available; if there are food boni, the calculation for the "smaller town" is even more unfavorable), and assume all tiles give 2gpt. Also assume the town is on the river, so we can run the cycle at 5-7. (Otherwise the result for the "larger town" is slightly smaller, as we need to run a 4-6 cycle.)
The small town runs 10 turns at size 1, 10 turns at size 2 and then spits out the settler when reaching size 3. So our total commerce/research is: 10 x 4 + 10 x 6 = 100
The big town runs 10 turns at size 5, 10 turns at size 6 and then spits out the settler when reaching size 7. So our total commerce/research is: 10 x 12 + 10 x 14 = 260
That is a pretty steep price that you have to pay for a slower expansion...

In addition: the small town will during that time probably just manage to get the required 30 shields for the settler. The big town can first build 2-3 warriors, before having to start collecting the shields for the settler.