The most common approach is what Dancy calls 'absolute objectification'. Its identifying characteristic is abstraction, as it peels away "mere appearances" to get at the Objective Reality that lies beneath. This is the objectivity sought by scientific reductionists, in pursuit of an 'absolute conception of reality'. One might think of it as a 'negative' sort of objectivity, as it is achieved by removing any subjectivity from the picture.
Dancy's 'Hegelian objectivity' is very different. It adds to the picture rather than taking away. It aims not at a 'view from nowhere', but rather a view from everywhere. The metaphor of concentric circles is illustrative: Think of your subjective viewpoint as the central circle. Let each successive circle represent a new observer, who can see each earlier person - including their faults - and can therefore correct for some of their biases and so come to attain a slightly more objective view of the world. But he in turn will have flaws that are noticed by the next observer, etc. ad infinitum.
Note how much more inclusive this picture of objectivity is. As Dancy (p.147) puts it:
In this process of objectification, nothing is left behind. Every aspect of each succeeding view is retained, though maybe somewhat altered, in each of its successors.
This allows us to retain secondary qualities such as colour, beauty, and perhaps morality, in our objective conceptions of reality. Plus all those everyday objects (e.g. tables and chairs) that aren't fundamental particles of physics and so might not survive a full-scale scientific reductionism.
I'm fairly inclined towards reductionism myself (see the earlier link). I see no reason to think that the 'noumenal realm' or ultimate reality will be anything like the 'phenomenal realm' of our common-sense conception. So if we're seeking Truth, then absolute objectivity could well be the way to go. But it seems that Hegelian objectivity is more likely to lead to understanding. In part, this is because an absolute 'view from nowhere' is so utterly inconceivable. Just ask yourself, how does reality "really" look? One thing I loved about The Matrix was towards the end of the movie when Neo suddenly 'saw through' the "appearances" of the Matrix, and everything turned into streams of numbers. That's my metaphor for trying to 'see' from the absolute conception: when the appearances are stripped away, what's left? What would it look like - how would it appear? There is, of course, no answer; it's a senseless question. But such confusions are what we find by attempting to grasp the world through absolute objectification. Given our human limitations, it seems that the Hegelian approach will tend to be much more illuminating to us.
That's my current position, anyway. What do you think? Which style of objectivity is better? Or do they just serve different purposes? Are the objects of a Hegelian view properly said to be "real", or should that status be reserved for the absolute conception?