Original question is similar to a lot of those put on this site (which I happen to think is a great site). What is the similarity? The lack of details in the original question. To achieve valid and succinct replies, the original question should have included the type of soil being compacted, the type of fill structure - e.g., under foundation, general backfill, water dam, etc. Then, respondents can give solid points to a specific question.
I always suggest to contractors to compact a bit on the dry side of OMC as energy of most compactors is higher than that of the laboratory proctors (modified (heavy) or standard (light)) unless there is solid need to do otherwise (e.g., dam core). I do not suggest that such be 3 to 4 percent less, but somewhat less. If there is no compelling need to add water, don't. Most granular fills can achieve compaction on the dry side. What I have noted for several years now is that with the placement of WMM (Water mixed macadam - a well graded crushed granular material - such as Ontario Granular A - typically used for base courses), if the OMC is exceeded, there is a great chance of noting pumping/sponginess during compaction with tell-tale wheel paths and cracking between dual wheels. Why? I suggest that by being over OMC, with vibration, the forces cause excess water to rise and carry with it the finest materials of the mix. The water evaporates leaving a thin film of fine silt over the top. With further compactive effort, additional water will rise but will not be able to escape through the low permeability silt skin. This leads to short term sponginess (which clients, if they see, get paranoid about). On drying, things will be okay. The biggest problem seems to be (when there is a water truck available) that contractors actually provide no control on amount of water to add. Some simply leave it to the truck driver to do it . . . whenever. As a result, I have seen good granular bases and subbases turn to sponges.
With clays - if you want strength, go slight dry of optimum - if you need to achieve low permeability, go wet. Kneed the clay. If clay saturates after compaction, strength (e.g., CBR) reduces. But this should all be taken into design. Clays need careful attention.
Comment on tailoring specification to the job is very important - off shelf specs are the scourge of the constructing engineers. For instance, if you have a 4m embankment of soil being placed atop 6m of soft to firm clay that will consolidate significantly, what is the sense of specifying 95% heavy compaction (modified)? No 4m embankment will fail due to nominal compaction done in a professional manner - settlement will take place; but is is not caused by "low" compaction.
Keep up the chats on topics such as this. They, as one put it, are not trivial. Compaction is an art, still. What really gets to me though, is that EVERYONE (read that clients, lay people, etc.) are all experts in such when most don't know more than a buzz word or so.
Best regards to all.