As the previous poster stated, diodes conduct at all voltages. The I vs. V curve goes continuously through the origin (0,0) with a finite slope.
In the forward bias direction, the current increases exponentially (very fast) with voltage, so there is no well-defined "knee".
In the reverse bias direction, as the voltage goes negative the negative current saturates at a low "leakage" level, but increases abruptly at a well-defined knee voltage, becoming a Zener diode.
For on-purpose "Zener" diodes with knees greater than about 5 V, the diodes are actually "avalanche" diodes and the current increases much faster above the knee than with true Zener diodes at, say, 3.3 V knee.
By the way, if you are measuring current for a diode in a glass case (e.g., 1N4148), you may see current at zero voltage due to photocurrent induced by room lighting.