To date I have reviewed hundreds of thousands of pages of various 19th century census records.
These records span most of the century and cover several hundreds of places in modern day Belarus, Ukraine, Latvia and Lithuania.
People with no last names are mostly visible in the earlier censuses 1811 and 1816. However as the rule, even during this time there are very few of them in any given town (with minor exceptions).
In addition some of these individuals recorded with no family either died, or were out of town for any reason at the time of the census. Thus the census takers often had no means of retrieving their last name.
In my attempt to index family names I have recorded those individuals based on their father's name. Thus for example Leizer son of Yankel will appear on this blog in the form of Unknown/Yankelovich.
Hopefully this will help researchers who have reached this time period but cannot find their family names in the given census to further explore such families and either find their ancestors by additional combination of facts
where applicable or fully rule out their presence in the given town/census/year.