The Ten Commandments are traditionally divided into two "tablets," the former concerning the worship of God, the latter concerning human society. Let us begin with the latter, as it easier to argue for them as foundational to American law, whereas the former must face the objection that they are unenforceable on account of the Establishment Clause of the First Amendment: "Congress shall make no law respecting an establishment of religion."
The Commandments of the latter tablet are (Note that for the sake of this article I will use the enumeration of the Commandments traditional in Orthodox, Anglican, and Reformed churches, following the King James Version of Exod. 20:1-21, the Ten Commandments being found again in almost identical form in Deut. 5:6-21.):
5. Honor thy father and thy mother.6. Thou shalt not kill.7. Thou shalt not commit adultery.8. Thou shalt not steal.9. Thou shalt not bear false witness against thy neighbor.10. Thou shalt not covet thy neighbor's house, [nor] thy neighbor's wife, . . . nor any thing that is thy neighbor's.
In passing, I would like to urge here that everyone, especially politicians, who believes that the Ten Commandments should be displayed proudly in schools, courthouses, etc., should memorize the Ten Commandments and be able to recite them easily to anyone who asks. Otherwise that person can readily be made to look like a Pharisee. More broadly, I personally think that all Christians and Jews should know them by heart.
With the exception of the Tenth Commandment, in the Old Testament these are enforceable laws, not just moral precepts to be adopted only by those with faith. Elsewhere in the Pentateuch more specific laws are given that include severe legal penalties for breaking Commandments. As for the Fifth Commandment:
If a man have a stubborn and rebellious son, which will not obey the voice of his father, or the voice of his mother, and that, when they have chastened him, will not hearken unto them: Then shall his father and his mother lay hold on him, and bring him out unto the elders of his city, and unto the gate of his place; And they shall say unto the elders of his city, This our son is stubborn and rebellious, he will not obey our voice; he is a glutton, and a drunkard. And all the men of his city shall stone him with stones, that he die: so shalt thou put evil away from among you; and all Israel shall hear, and fear. (Deut. 21:18-21)
As for the Sixth:
But if any man hate his neighbor, and lie in wait for him, and rise up against him, and smite him mortally that he die, and fleeth into one of these cities: Then the elders of his city shall send and fetch him thence, and deliver him into the hand of the avenger of blood, that he may die. (Deut. 19:11-12)
As for the Seventh:
And the man that committeth adultery with another man's wife, even he that committeth adultery with his neighbor's wife, the adulterer and the adulteress shall surely be put to death. (Lev. 20:10)
As for the Eighth and Ninth (which concerns swearing falsely, not every act of dishonesty, as Osler claims):
If a soul sin, and commit a trespass against the LORD, and lie unto his neighbour in that which was delivered him to keep, or in fellowship, or in a thing taken away by violence, or hath deceived his neighbor; Or have found that which was lost, and lieth concerning it, and sweareth falsely; in any of all these that a man doeth, sinning therein: Then it shall be, because he hath sinned, and is guilty, that he shall restore that which he took violently away, or the thing which he hath deceitfully gotten, or that which was delivered him to keep, or the lost thing which he found, Or all that about which he hath sworn falsely; he shall even restore it in the principal, and shall add the fifth part more thereto, and give it unto him to whom it appertaineth, in the day of his trespass offering. (Lev. 6:2-5)
The Tenth Commandment, by its very nature, prohibits those inner feelings and vices that lead people to break the Sixth through Ninth Commandments. As such, it pertains to morality rather than to civil law. I am not aware of any law in the Old Testament that attempts to legislate against coveting. This, however, is not an argument in Osler's behalf since the Tenth Commandment was never intended to be a matter of civil law.
This being said, there have been laws in America that line up with the Fifth through Ninth Commandments. Osler argues that there is no law against lying, but that is a misinterpretation of the Ninth Commandment, which prohibits swearing falsely, which is also prohibited by American law. As for the Seventh, even he admits that there are laws against adultery in America, though they have lapsed into desuetude. It occurs to me, though, that the conservative Christian argument would precisely be that a return to the Ten Commandments is needed at this time because for the last half-century society has been descending into immorality, including, certainly, adultery. The issue is not whether the Seventh Commandment has fallen into desuetude (for they would admit that it has!) in American law, but whether it is a part of American law, which Osler admits it is. As for the Sixth and Eighth Commandments, he readily concedes that both theft and murder are prohibited by American law, though he points out that the Bill of Rights guarantees due process, a speedy trial, etc. But there's nothing in the Ten Commandments that says those accused of breaking these Commandments shouldn't receive a fair trial and so forth (and I think any reasonable-minded conservative Christian would agree that they should and that this is no violation of the Ten Commandments), so this is beside the point.
The point that most contradicts the conservative Christian argument is probably the interpretation of the Fifth Commandment. Probably examples of punishing children for disobedience can be found somewhere in American law, though surely such laws have long since lapsed into desuetude, and I suspect that few conservative Christians would wish to revive any such laws that might be found on the books somewhere. Fundamentally, of course, this Commandment is enforced in the home, as would also have been the case in ancient Israel, though as I showed, in an extreme case it could become a capital offense. I suppose that a conservative Christian could argue that this Commandment, in light of modern circumstances, could be left to the private domain, but that already would seem to be an argument in favor of Osler's reading of the Ten Commandments as having a purely private, religious role in American society, rather than a civil, legal role.
I'll leave the former tablet aside for now because it would be difficult to argue that it can be legislated. Certainly the interpretation that the Establishment Clause has been given over many decades of American jurisprudence precludes enforcing the former tablet. There are, of course, conservative Christians who argue that that should merely prohibit establishing a particular sect as the official religion. This is consistent with the way the Establishment Clause was once interpreted, as laws against both blasphemy and conducting certain business on Sundays are to be found in American law.