In the world we're discussing, all three are infectious and all three begin the transformation with a single bite.
The Vampire is a 'traditional' vampire - undead, no soul, no reflection, feeds on blood. Vulnerable to sunlight, wooden stakes in the heart, and poorly written romances. Bitten humans die after being drained of blood, rise on the next sunset as a new vampire.
The Werewolf is a 'traditional' werewolf - living, possesses a soul, transforms into a wolf (either fully, or into some sort of wolf-man state) uncontrollably during the three nights of the full moon each month, lacks control of actions while transformed. Vulnerable to silver (and Nair?). Bitten (but not killed) humans live as normal, until the next set of full moons, at which point they transform.
The zombie is a 'Romero' style or one of it's derivatives - undead, hungry, no discernible intellect, no healing ability, driven to consume living flesh (human preferred). Vulnerable to physical damage and especially brain destruction. Bitten humans die within a short time (typically less than 24 hours) and rise shortly thereafter as new zombies.
There are 6 possible orders (V = vampire, Z = Zombie, W = werewolf):
Four of these should be fairly straightforward, and I'll handle these first.
ZVW, ZWV: The zombie bites a human. Then (presumably before s/he turns) the human is bitten by a werewolf and vampire (order unimportant in this case). Even assuming the zombie bite happens at night on the night of the full moon, I expect the human will expire and reanimate as a 'normal' zombie - all three transformations take time, and the zombie is simply the fastest. It kills and raises within 24 hours (typically), longer only if they receive significant medical aid. Unless they manage to last 48 hours (unlikely, if they receive additional wounds from more bites) the zombie virus kills them, then they get up to kill.
VWZ, VZW: Again, the speed of the transformation is important here. The only way all three bites are likely to occur on the same individual is if they all happen within a 48-hour period. Zombies wouldn't be attracted to the dead flesh of a vampire, and the bite of a werewolf has fairly consistently been shown to be either fatal or simply damaging to a vampire. It's never been seen to trigger some additional transformation (except in a Ghostbusters cartoon episode, and some poorly done movies). If the human lives long enough to become a vampire, they die and rise the next night as a vampire. If the human receives a zombie bite just before (or right after) expiring, they will still likely rise a vampire. If, however, the human is bitten by the zombie early enough before death, the quicker zombie virus will claim his life and raise him as a zombie.
WZV: Zombies would be drawn to the living flesh of a werewolf. After a zombie bite, it's likely the werewolf would die and rise as a zombie. If he died while still in wolf or hybrid form, he would become an undead version of that form. Compared to the zombie bite, the vampire bite does not act quickly enough to challenge this transformation.
WVZ: Here's where the fun begins. It's never been seen for a werewolf (in wolf form) to be infected by a vampire bite - traditionally vampire bites simply kill werewolves. Thus, it's only likely that a wolf would be vampire-bitten (with any effect other than death) while in human form. If it did not simply kill him, I expect he would rise as a normal vampire - vampire's bodies have been consistently shown to be unchanging, often restored to their state at death at the beginning of each night. Therefore, it's doubtful that the wolf transformation would occur. Assuming the zombie bite happened with enough time remaining before the vampire reanimation, the quicker zombie virus would raise the human as a zombie. If not, they would become a vampire.
Essentially, the genres don't mix well. A zombified wolfman would still have his sense of smell, hearing, and eyesight, but would move like a zombie. It's likely his other senses would fade as he decomposed. The other 'mixes' (zombie vampire, wolfpire) are highly unlikely to occur.
White Wolf, which writes games that allow you to play as werewolves, vampires, etc addresses this issue directly: Vampires can't become any of the other 'races', werewolves can't be turned into vampires (and the vampire who drinks their blood regrets it), etc.