Theories on how the Saiyans' tails disappeared:

Gohan (Tail apparently disappeared after Vegeta cut it off)

Theory #1:

*After Vegeta is defeated and Goku is hospitalized.

One random day before Gohan flies off to Namek:

Gohan: Hey look, my tail finally regrew back!

Goku: That tail causes us Saiyans to transform into giant apes when the moon is full and it was the cause of your grandfather, Gohan's death. On top of that, we are prone to losing control and cause great destruction when we transform.

Chi Chi: OMG I can't allow our son to become an ugly ape! It'll get in the way of his studies!

*Uses "Kaio-ken" and ends Gohan's tail permanently*

Vegeta (Tail apparently disappeared after Yarijobe cut if off)

Theory #2:

*After Frieza is defeated and Vegeta is staying in Capsule Corp.

One random day before Goku returns back to Earth:

Yamcha: Hey! Your tail's back!

Vegeta: About time.

Bulma: You should get rid of it.

Vegeta: Give me one reason why.

Bulma: I'll give you 2.

1. Your tail's uselesss without the Moon.

2. Goku. Has no tail. Still way stronger. Than you.

Vegeta: ...

Bulma: Deep down, you know you can never truly measure up to him with that tail

Vegeta: ... ... ... Dammn you, Kakarottttttttttt-!

*Flies to Kami and ends tail permanently*

(Back at Capsule Corp)

Yamcha: Great job, Bulma! Never knew you were such a master manipulator!

Bulma: Oh, I still have some tricks up my sleeve to bend him to my will.. Snigger snigger

*Some time later..

Vegeta wears pink shirt and yellow pants against his will.

Trunks (Apparently born without a tail)

Theory #3:

*After birth, where everyone is still training in preparation for the Andriods.

Piccolo: Hey Bulma.

Bulma: Hey..Piccolo.. What a rare visit..

Piccolo: Is Trunks born yet?

Bulma: Wha-

Piccolo:Oh wait, there he is. With a tail. As expected.

Bulma: Huh-

Piccolo: I need to borrow him for awhile. -Yoink-

Bulma: Wher-

*Flies off somewhere and return back some time later

Piccolo: All yours, good as new.

*Flies off.

Bulma: Heyyy-

*Gets distracted by Vegeta's return

Bulma: Vegeta! You're back! I know you're busy training and all to notice, but here's your son.

Vegeta: WHAT? BULLS%^+! HE DOESN'T HAVE A TAIL!

Goten (Also apparently born without a tail)

Theory#4:

*After Goku is dead.

Gohan(Still a kid): Look ma, little Goten has a tail too! Just like

me when I was young.

Chi chi: Hmm, I think I'll let this slide. I feel guilty about Goku

being dead partly because I was too strict with you and all..

* A few seconds later

Chi chi: Gohan! What happened to the house? I was only out for a few seconds!

Gohan: Goten's tail did it!

Chi chi: Okay, that's it. Off with his tail!

*End of Goten's tail*