My Knight is level 7 and still only ever deals 1 attack per round, at which level does he start to get 2 attacks?

And what about the Paladin and Archer?

I don't remember exactly, but I think you get a second attack at level 8. I'm pretty sure the Paladin and Archer do as well (and this includes the archer's bow attacks).

Interesting.

Heck, I hope my thief get extra attacks eventually too. He really hasn't been all that useful in battle so far.

"Attacks" is probably not what you think it is. The round order never changes, it's just that after a certain number of levels, you gain more "attacks" per round.
Suppose you have a 1d4 weapon. It will do between 1 and 4 damage every time you attack with it. When you get 2 attacks, it will do between 2 and 8 damage. When you get 3 attacks, it will do between 3 and 24 damage. For each "attack" you get, when you press the attack button, your character will effectively attack that many times. There's no indication that additional attacks are going on other than you'll be doing more damage (which you can gauge with the on screen blood splat)
Knights and ninjas gain another attack every 5 levels; paladins, archers, robbers, and rangers gain another attack every 6 levels; clerics and druids gain another attack every 7 levels; sorcerers gain another attack every 8 levels; barbarians gain a new attack after only 4 levels, making them one of the strongest characters in the game at the end (L200/4 * 50 for an obsidian weapon = 2500 damage per round)

Ack! wait, you said MM1. I was in World of Xeen thinking. Ignore my post, it doesn't apply to MM1 at all.

I think in most MM1 versions fighter, paladin and archer get an extra attack per round every 8 levels, while thief, cleric and mage stay at one attack per round.

Although I don't remember what level each class must be for multiple attacks, I do recall that the game specifically uses the syntax "Knight attacked X times & hit X times for XX damage" when you attack.

I can confirm that my cleric, mage and robber never got any extra attacks whereas my other characters were up to 3 attacks per round when I finished the game.

Both my Paladin and my Knight go up to two attacks (from one) when I hit them with Superheroism, which gives them three effective levels. My Knight is at level seven, and my Paladin at level six - so (assuming that they get the extra attack at the same level) they get extra attacks either at level eight or nine.
Probably eight. Eight is a happy number for RPGs.

I'm pretty sure it's eight. I also think this info is in the manual somewhere.