First of all, there are seven distinct traditions which divide the seven verses in Exodus differently regarding the numbering of the so-called ten commandments.
Almost all (6 of7) agree that the first is "You Shall Have No Other Gods Before Me", one makes it the second, counting the prologue "I am the Lord your God" as the first
Some (4 of 7) count "You Shall Not Have False Idols" as the second, two other combine this with the first and call it the first
The next few have 4 of 7 agreeing, while the other 3 are one step behind:
"You Shall Not Take the Name of the Lord in Vain" is third/second
"Remember the Sabbath Day to Keep it Holy" is the fourth/third
"Honor Thy Father and Mother" is the fifth/fourth
"Thou Shalt Not Kill" is the sixth/fifth, although one version counts it as the seventh commandment
"Thou Shalt Not Commit Adultery" is the seventh/sixth, although since one source reverses this and the previous command, it is 3 for seventh and 4 for sixth
"Thou Shalt Not Steal" is eighth/seventh
"Thou Shalt Not Bear False Witness (Lie?) Against Thy Neighbor" is ninth/eighth
"Thou Shalt Not Covet Thy Neighbor's House" is 6 for the tenth (some combined with the following) and 1 for the ninth
"Thou Shalt Not Covet Thy Neighbor's Wife" 5 of 7 combine it with the previous for tenth and two call it the ninth
"Thou Shalt Not Covet Thy Neighbor's Stuff" - all call it the tenth combined with either one or two of the previous
(4 of 7 combine all 3 "covets" into #10, 3 combine 2 of them in various ways)
Even though there are actually thirteen commandments, if you count every one that at least one tradition considers a separate commandment, the bible, just before the listing, specifically calls them ten commandments (or words, sayings, or matters), but does not clearly delineate where one "saying" ends and another begins. Maybe whoever wrote it thought it would be obvious, or that it was unimportant. Here they are separated out and listed in order:
- I am the Lord your God
- You Shall Have No Other Gods Before Me
- You Shall Not Have False Idols
- You Shall Not Take the Name of the Lord in Vain
- Remember the Sabbath Day to Keep it Holy
- Honor Thy Father and Mother
- Thou Shalt Not Kill
- Thou Shalt Not Commit Adultery
- Thou Shalt Not Steal
- Thou Shalt Not Bear False Witness (Lie?) Against Thy Neighbor
- Thou Shalt Not Covet Thy Neighbor's House
- Thou Shalt Not Covet Thy Neighbor's Wife
- Thou Shalt Not Covet Thy Neighbor's Stuff
Periodically there are legal disputes about posting the Ten Commandments in courthouses and in government buildings. The rationale being that the commandments are all basic moral/ethical stuff and no one should have any problem with them, even if hey did originate in a religious book. There are some problems with that position. Some of them are hard core religious and not just "do unto others..." stuff.
The first three to five, depending on how you're counting, address who you should be worshipping, when you should do it, and how you should talk about the god who is the object of this worship. For anyone who worships a god different from the god of the bible, this is not something that they they would want to do and it certainly shouldn't be displayed in a government setting. If you think that the whole world, or at least the whole country, consists entirely of Christians and Jews (and depending on how you view Allah, Muslims), then you might have no problem with this. But that's not the reality - there's a large number of people who worship or honor different gods (or no gods at all) - in addition there's that pesky First Amendment.
The "covet" prohibitions are also problematic. Most people would agree in theory that these are things that you shouldn't do, but when you get down to it they are prohibitions against thought not actions. So once again, this comes down to strictly religious rules, not anything that could or should be encoded into secular law.
Honoring your parents (unless your parents are evil bastards), no stealing, lying, killing or cheating on your spouse...I'll give you those. The Five Commandments.