I don't believe I've ever seen Goten or Trunks with tails, which doesn't quite make sense, seeing as how Gohan had a tail at his birth, and was also half saiyan half human. From what I've read, it's assumed that they were cut off at birth, but I'd like to know officially if that's the case, since the articles I read had no sources. And I'm pretty sure Saiyan tails normally grow back. Is there anywhere in DBZ that they mention the permanent removal of Goten's and Trunks's tails?
A common theory is that Akira Toriyama just didn't want to / forgot that Saiyans had tails. This wouldn't be surprising as Akira makes several other errors/omittances in the series.
Here's an interview segment from him talking about Goku's tail:
It’s true. (laughs)
Goku was originally an actual monkey in the initial sketches, after all. My editor told me, “Without a tail, he has no distinguishing traits,” so I added a tail.
That tail I added was such a nuisance when I was drawing, I couldn’t stand it… so I immediately thought up an episode where his tail gets cut off. (laughs)
It wouldn't be surprising if having gotten rid of the tail that he so hated, he didn't really want to reintroduce it into the story again, so rather he left Trunks and Gohen without one.
There is very little by way of actual evidence to answer this question (for example, in the manga Krillin asks Bulma why Trunks has no tail, and whether she had it removed; she never gives an answer). There is however a lot of speculation on the internets about it.
The closest thing to an official answer can be found in the "DragonBall Daizenshuu 4 – World Guide" databook (1995, Shueisha Inc.), endorsed by Toriyama Akira, which has this to say (on the subject of "Saiyan"):
The tailless second generation are super ultra child prodigies.
Saiyan genes have an extraordinarily good compatibility with Earthling blood. Because of this, when the two races are mixed together children with formidable power are born. Particularly, those Halflings born without tails hide an exceptional battle power. There are many things that they naturally master from a young age, such as the ordinarily arduous transformation into a Super Saiyan. In spite of having such an outstanding battle sense, they do not have a fondness for battle like a pure Saiyan. Instead, it seems that the violent temperament of the Saiyan has been relaxed through their Earthling blood.
The text doesn't outright say it, but the strong implication is that a Saiyan/human hybrid is not born with a tail (or at the very least, some of them are born without tails, and those of this category are notable for being extraordinarily powerful).