Slow blow or fast acting fuses are fair enough. Fuses  meant to protect the devices when excessive amount of current drawn to your circuitry beyond the calculated value or input current specification. Tolerances can be determined according to the type of application you will have, inductive load, capacitive load or seemingly linear.
The degree of protection depends on your choices.

I believe it is best to provide as much protection as you can by using larger number of fuses with less rating.  The optimal design would be to have a fused terminal block with fast acting fuses with a blown fuse indicator to allow for easy troubleshooting.  By minimizing the number of fuses and bundling more devices under larger fuses you run the risk of failure to your devices.  When you use the term control circuit it lends itself to a large spectrum of possibilities.  Does your control circuit consist of Semiconductor Protection (Extremely fast acting), Metering circuits with relays (Fast Acting), General Purpose 125VAC circuits (Medium Lag).  I would recommend not using time delay fuses for any type of controls.

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Jacobs Engineering Group Inc.
1041 East Butler Road
Post Office Box 5456
Greenville South Carolina  29606-5456
864·676·6458 Phone  864·676·4789 Fax
E-mail: jason.buda@jacobs.com

Jason W. Buda
Electrical and Control Systems Department
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Suggestions marked ///\\\:
What type of fuses should I use on the AC and DC of my automation
control circuitry (1A, 3A, 10A)?
///It depends on the circuit load. The lowest fuse rating, which does not trip, is the best since it protects the circuit most sensitively. The current may be obtained by the clamp-on ammeter, and fuse rating may be as low as 1.5 x Imeasured for dual element (time delay) fuse, or 3 x Imeasured for fast acting fuse. Make sure that the circuits are fully loaded. Caution shall be exercised when it comes to wire sizes. Less then #18 AWG is not recommended.\\\
 Are there many options with fuses or simply fast-acting vs. time-delay?
///Perhaps, you could use resettable fuses or circuit breakers.  There are however different types of fuses according to their time-current curves. There are also low peak fuses, which protect circuit and equipment from short circuit damages most.\\\