In the circuit below
Do I need to use a plane for AGND.
The control signals are from a ULN2003 Darlington Transistors

 

You probably do not need a ground plane, but it can be good practice to design the board so the GND currents of each output only meet in one place. A ground plane with slots in it can work well.
The common GND point would be the -ve terminal of your main power filter cap, that looks to be missing from your circuit. It is always worth while adding power filtering and bypassing to keep the system power noise down.

 

You probably do not need a ground plane, but it can be good practice to design the board so the GND currents of each output only meet in one place. A ground plane with slots in it can work well.
The common GND point would be the -ve terminal of your main power filter cap, that looks to be missing from your circuit. It is always worthwhile adding power filtering and bypassing to keep the system power noise down.

The thing to be noted is that all the control signals are from another board through a Darlington transistor. The +12V on the input side of the optocoupler VOD205T is also from the same board and each control signal is connected to the collector of the Darlington transistor. The LED will turn on only when a +ve voltage is applied to the base of each transistor.
So my doubt is do I need a Ground plane in the input side of the optocoupler?

 

It depends on the nature of the signal and to what extent if any noise is a concern. If these are digital signals at "normal CMOS levels" (whatever that may be) then you probably don't need a ground plane.

Edit: (Schematic finally downloaded) Those look like on/off control signals. I would not bother with a ground plane unless there are other circuits that could be affected this one's operation.

 

It depends on the nature of the signal and to what extent if any noise is a concern. If these are digital signals at "normal CMOS levels" (whatever that may be) then you probably don't need a ground plane.

Edit: (Schematic finally downloaded) Those look like on/off control signals. I would not bother with a ground plane unless there are other circuits that could be affected this one's operation.

These are on/off control signals.
I am using ULN2003 Darlington transistor to drive the optocoupler.
Is 3.3V from a controller is sufficient to switch ULN2003 ?.

 

I'm not intimately familiar with Eagle, so my question is whether +12V with an arrow point is the same net as +12 with a circle? If so, the use of the optocoupler would be moot.

Assuming they are separate, you do not need a ground plane. However, best practice would be to have the +12V with an arrow trace follow each of the CTRL signals to the optocoupler as closely as possible. You want to minimize the space between the CTRL signal and the return current path, in this case the +12V with an arrow. Either run a +12V with an arrow trace adjacent to each of the CTRL signals or on the opposite side of the board to them until it reaches the current limiting resistor. Place each of the current limiting resistors as close as you can to the optocoupler.

Think of it this way: You have a circuit with a voltage source, +12V, a current limiting resistor and an LED with the cathode of the LED connected to your voltage return or reference. Since there is space between each of these components, this circuit will form a loop. Being a loop, it will be subject to the influence of electromagnetic fields passing inside the loop. Effectively, it's an antenna. If it's big enough and if the EM field is strong enough, it's possible this antenna will pick up enough energy to trigger your LED.

With antennas, size means a lot. The larger the loop, the lower the frequencies that the antenna will receive. Fortunately for you, your parts are small so you won't have any reason to create a large loop and a loop that small won't be susceptible to the lower frequencies you'll be working with.

 

These are on/off control signals.
I am using ULN2003 Darlington transistor to drive the optocoupler.
Is 3.3V from a controller is sufficient to switch ULN2003 ?.

Since you don't need much current out the ULN2003 should work fine.