On Yom Kippur and Tisha B'av we read about the Ten Martyrs who were gathered and killed by Emperor Hadrian to atone for the Sale of Yosef HaTzadik.
In addition to the idea that the Ten Martyrs didn't live at the same time, the Talmud records the deaths of at least some of the Ten Martyrs in a very different manner (I remember learning about the deaths of some of the Rabbis, and assume that the rest are also recorded). According to the Talmud each Rabbi was individually found guilty of a crime (such as teaching Torah or giving Semicha) and executed.
So, if we assume that the poet was taking some poetic license and the deaths of the Ten Martyrs didn't happen exactly as described (i.e. they weren't all gathered together and then killed at one time). How did they actually die?
Where in the Talmud (or other sources from the same time period) are those deaths recorded and what crime did the Rabbis commit? How were they actually killed? If there are conflicting accounts in other places, please mention them.
I'm setting up a community wiki answer in order to try and get a complete answer in one place.