The Handstand is one of Lara's fancy moves that has no real use in the games, except that it looks pretty.
The Angel of Darkness
The same works in Angel of Darkness.
Tomb Raider Legend
To perform a handstand in Tomb Raider Legend simply hold the jump button a little longer.
Tomb Raider Underworld
To perform a handstand in Tomb Raider Underworld you need to press and hold the Fire Weapon(s) key while pressing jump and up to mount a ledge.
- PC: H or Left Mouse Button for fire, Space for jump.
- PS3: R2 for fire, x for jump.
- Xbox 360: RT for fire, A for jump.
Reasons might be found on the "discussion page".