Power supply of 5V 1A causing erratic behavior of Tower Pro SG90,- what's the reason for this? Also occurs when using the MB102
The MB102 has a voltage regulator.
The 5V USB &quot;charger&quot; has a voltage regulator.

No, you don't need to add a regulator to either of those two.
Since you mention the MB102, it seems you have everything wired to a breadboard.
That's generally not a good idea.  Breadboard contacts are not made for high current - the few hundred milliamperes that the SG90 will draw count as &quot;high current&quot; when working with a breadboard.
Beyond that, I can tell you that a 5V, 1A power supply should be more than adequate for the SG90.  I've built gadgets that used a single 5V 1A power supply for multiple SG90 servos.  The current wasn't a problem.
I have a gadget sitting here close by that uses a 1.5A power supply for two much larger servos (20kg cm rather than the 2.5kg cm of the SG90) that routinely move a heavy object.

Make sure that you have a good ground connection between the servo and the Arduino.  The PWM signal needs the ground, especially if you are powering the Arduino from a power supply separate from the servo power supply.
Wire your circuit without using a breadboard.  That means soldered connections and proper connectors between the components.
If the servo is still erratic, you could add a large capacitor to the power leads, in parallel to the power supply.
If it is still erratic, check the power supply voltage while the servo is operating.  It should stay at 5V, even when the servo moves.  If the voltage drops, then you need a better power supply.