The following will be a growing document. I will add to it every time I encounter a new argument.
Intro
"From a point of view outside of this affair, the killing of a neurologically inactive fetus is no greater a harm than the killing of a mouse, and in fact decidedly less--a mouse is neurologically active, and though it lacks a complex cerebral cortex, it has a brain of suitable complexity to perceive pain." ~ Dr. Richard C. Carrier
The most frequent canard I encounter in discussing the morality of abortion is that it takes a life. As with any philosophical discussion, we need to first define our terms to ensure that we are all on the same page when we use the word "life" (or the words "baby" and "child" for that matter). After all, many things have life; they possess cells that replicate, undergo the Krebs Cycle, etc. Objects on this list include trees, sperm, and cancerous tumors. The list is literally so extensive as to defy concise description. So obviously "life" in the sense of merely being alive doesn't mean all that much, since we typically don't weep over chopping down a tree and we can be downright insistent on purging tumors. Clearly, there is a difference between something being alive and something having a life, and it seems that those who are opposed to abortion also make this distinction whether they admit it or not, since none of them mourn the loss of a tree.
Where then, do we draw the line on when it is alright to destroy something that is living (because if the loss of living cells bothers you, it's time to stop scratching your nose)? This point must be reached somewhere, since most of us would consider it morally wicked to snuff out the life of a freshly born human being. What are the criteria? At what point does the collection of cells within the pregnant female acquire a "life" such that it is separated in terms of humanity from other living things?
I suggest it has everything to do with consciousness and the ability of a being to suffer its own loss. This even makes sense medically. The point at which you are medically dead is when there is no activity in your central cortex (when your brain "flat lines"). At this point your heart may still be beating with the help of a pacemaker, your cells will still be active, and your lungs may still be breathing with the help of a respirator, but this is all irrelevant since the conscience that constituted "you" is gone and will never return. At that point your body is simply engaging in autonomous functions. There is no reason to consider you still alive or to grieve when we unplug your life support.
With a fetus, the brain does not become electrically active until the fifth month (typically the 20th week). Until that point, it is no different than killing anything else that is alive but cannot suffer its own loss. When we say the words "baby" or "child", we imply a creature that has lost something when it stops "living"; a zygote, an embryo, or a fetus is not a child. It is a collection of cells (until the end of the second month when minute organ development begins to occur), nothing more.
Argument from potentialAn argument that almost unavoidably arises at this point is that a zygote will one day become a child (perhaps the next Beethoven!) if left unchecked. These people do not seem to realize that every sperm in my body is a potential human being (it just needs the female egg, itself a potential unique, glorious human being). Yet the prospect of this lost potential does not seem to frighten them out of their chastity (trust me, it doesn't). Why not?
I am certain that if all human beings went Sodom and Gomorrah-style crazy with lust, we would eventually produce a physicist that would dwarf Stephen Hawking or Einstein, or a composer that would reduce the work of Mozart to child's play. That's no reason to do it. The dire consequences of augmented, unmitigated population growth are very well documented. A woman should feel no obligation to produce a child she does not want, simply because that kid
might be the first John Conner.
Why then do we care about the potential of a particular set of cells in the female because they share a similar potential?
Argument about souls"Before I formed you in the womb I knew you, And before you were born I consecrated you; I have appointed you a prophet to the nations." ~ Jeremiah 1:5
There is not a scrap of evidence to suggest we have a soul that existed before birth or that will exist after death. All evidence suggests that conscience is tied exclusively to a functioning brain. Religious people can profess their certainty in this until they are blue in the face, but until they can evince even a decent reason to believe that it's true, they are just wrong by any rational standard.
Of course, nobody is insisting that religious people forfeit the right to reproduce as they please; nobody wishes abortion to be forced on anyone, even if their reasons for avoiding them are ludicrous. But you most certainly do not get to use your delusions to inflict others with their arbitrary, unfounded standard.
Argument from post-abortion traumaIn 1987, Ronald Regan asked then Surgeon General C. Everett Koop to produce a report assessing the possibility of trauma in the would-be mother. Scott's report concluded that "the available scientific evidence about the psychological sequelae of abortion simply cannot support either the preconceived notions of those pro-life or those pro-choice." This was as generous as the highly conservative Koop could be without being flagrantly dishonest. However, Regan ordered the report to be re-written to say that women did suffer post-abortion trauma, and it was (though Koop separated himself from it both publicly and during investiation) from this re-written report that the idea that abortions induced trauma was born.

Since both the announcement of that one disingenuous paper and the resulting beat down from the scientific and medical communities, all of our credible medical bodies have been in perfect harmony on the facts. The American Psychological Association
sums it up nicely:
"The most methodologically sound research indicates that among women who have a single, legal, first-trimester abortion of an unplanned pregnancy for nontherapeutic reasons, the relative risks of mental health problems are no greater than the risks among women who deliver an unplanned pregnancy."
All the same, even if a higher portion of women did suffer depression after their abortions, you cannot shackle everybody for that. Women should still get the choice. However, the science is conclusive that no greater threat to mental health exists for having an abortion, so the argument is moot.
OutroAbortion provides societal good by helping to mitigate population growth and by not saddling women with children they are not ready for, and there is no downside. All arguments to the contrary depend on either wrong information, outright lies, a lack of knowledge about the nature of pregnancy, beliefs about reality that are contra to the evidence and accepted on nothing more than faith, not spending enough time taking in the philosophy of the situation, or some conglomeration of all of the above. To quote Richard Carrier for the second time in this post:
"An act that causes no involuntary harm and produces some benefits for individuals and society in general should never be outlawed. This is based on the principle that laws should only exist to preserve and protect the liberty of individuals and, when no liberty is at stake either way, to increase the general welfare of all citizens."
There is no logical or empirical grounds upon which to illegalize abortions.