Since a single line doesn't do damage, it is possible to do
To achieve this, there are a couple of requirements:
To get these pieces one after the other
For this to occur so that the final piece also clears the board, we get these constraints on the number of pieces $X$:
Given these, the smallest $X$ that satisfies both requirements is
This is a small ...
It looks like
This works all nice and fine,
There may, of course, be something obvious I'm missing.
For the J:
With the L, there are some problems that are very similar to the Z case,
Here's the T:
And finally, the S:
And for completeness' sake, the solutions already found by others:
A simple solution involves
Stack the first five arranged the same direction (on the side of the T with the bottom facing either left or right. In my example, the bottom is facing the right). This eliminates the second row (shown with Xs through them). Next, flip the T 180 degrees so the bottom is facing the other direction and fill the gaps as such. This ...
Here is a link to a Mathematics and Computer Science journal published by Kaitlyn M. Tsudura from Saint Mary's University in Halifax, Canada. It summarizes the results of studies published by John Bruztowski and Heidi Burgiel
The studies were basically to determine whether it was possible to make a Tetris game only winnable but they found that it is ...