programmatically speaking, you can do both. the only difference between them (in code) is this :

 // using 00:00:00 will require you to add 1 day to the end date in order to count as full day
 if(time >= "2019-12-03 00:00:00" && time <= "2019-12-04 00:00:00")

//using 23:59:59 will not require you to add 1 day to the end date.
 if(time >= "2019-12-03 00:00:00" && time <= "2019-12-03 23:59:59")

so, basically, if you use 23:59:59 there is a one second off the grid, if any record has been stored in this second, it'll not be included in the results. while the second one will include it.

Which one to use ? surely the 00:00:00 if you want to be more precise, however, I've not seen any difference in the results in my projects as I've used both of them in different projects. But I'm sure there are some projects needs to include every micro second as this microsecond could change the result's curve (such as analytics or deep learning ..etc).

In SQL Server, don't save the time as string, save it with the correct datatype (DateTime, TimeSpan ..etc). SQL Server will reads the time perfectly fine when you pass a correspond time datatype from your application.

It depends on perspective:

var xx = DateTime.MaxValue.ToString("HH:mm:ss.fffffff");
var xy = DateTime.MinValue.ToString("HH:mm:ss.fffffff");

Gives

23:59:59.9999999

00:00:00.0000000

So, one is the end of the day and the other is the beginning of the day.

There's an interesting novel called 'The time between midnight'

DateTime.MaxValue is exactly that - the maximum value that DateTime can represent; that is to say, the 'last' point in a day. Conversely, the .Date property makes use of 00:00:00 by default, as it has no notion of time (unless specified).

If you have an event that occurs at exactly midnight, I would recommend storing it as 00:00:00, as the event occurs at midnight, and you want to accurately represent that.

Ultimately, it really depends on your desired use case as to which one you want to use. Do you want to state that the event occurs on day 1's evening, or day 2's beginning? That is what it boils down to, although in the vast majority of cases such a delineation makes no difference. In this case you would want to opt for both the accuracy and 'ease' of 00:00:00.