The Ten Commandments in Jewish Space versus the Public Square
וַיְדַבֵּר אֱלֹהִים אֵת כָּל הַדְּבָרִים הָאֵלֶּה לֵאמֹר. אָנֹכִי יְהוָה אֱלֹהֶיךָ אֲשֶׁר הוֹצֵאתִיךָ מֵאֶרֶץ מִצְרַיִם מִבֵּית עֲבָדִים לֹא יִהְיֶה לְךָ אֱלֹהִים אֲחֵרִים עַל פָּנָיַ. לֹא תַעֲשֶׂה לְךָ פֶסֶל וְכָל תְּמוּנָה אֲשֶׁר בַּשָּׁמַיִם מִמַּעַל וַאֲשֶׁר בָּאָרֶץ מִתַָּחַת וַאֲשֶׁר בַּמַּיִם מִתַּחַת לָאָרֶץ. לֹא תִשְׁתַּחְוֶה לָהֶם וְלֹא תָעָבְדֵם כִּי אָנֹכִי יְהוָה אֱלֹהֶיךָ אֵל קַנָּא פֹּקֵד עֲו‍ֹן אָבֹת עַל בָּנִים עַל שִׁלֵּשִׁים וְעַל רִבֵּעִים לְשֹׂנְאָי. וְעֹשֶׂה חֶסֶד לַאֲלָפִים לְאֹהֲבַי וּלְשֹׁמְרֵי מִצְו‍ֹתָי. לֹא תִשָּׂא אֶת שֵׁם יְהוָה אֱלֹהֶיךָ לַשָּׁוְא כִּי לֹא יְנַקֶּה יְהוָה אֵת אֲשֶׁר יִשָּׂא אֶת שְׁמוֹ לַשָּׁוְא. זָכוֹר אֶת יוֹם הַשַּׁבָּת לְקַדְּשׁוֹ. שֵׁשֶׁת יָמִים תַּעֲבֹד וְעָשִׂיתָ כָּל מְלַאכְתֶּךָ. וְיוֹם הַשְּׁבִיעִי שַׁבָּת לַיהוָה אֱלֹהֶיךָ לֹא תַעֲשֶׂה כָל מְלָאכָה אַתָּה וּבִנְךָ וּבִתֶּךָ עַבְדְּךָ וַאֲמָתְךָ וּבְהֶמְתֶּךָ וְגֵרְךָ אֲשֶׁר בִּשְׁעָרֶיךָ. כִּי שֵׁשֶׁת יָמִים עָשָׂה יְהוָה אֶת הַשָּׁמַיִם וְאֶת הָאָרֶץ אֶת הַיָּם וְאֶת כָּל אֲשֶׁר בָּם וַיָּנַח בַּיּוֹם הַשְּׁבִיעִי עַל כֵּן בֵּרַךְ יְהוָה אֶת יוֹם הַשַּׁבָּת וַיְקַדְּשֵׁהוּ. כַּבֵּד אֶת אָבִיךָ וְאֶת אִמֶּךָ לְמַעַן יַאֲרִכוּן יָמֶיךָ עַל הָאֲדָמָה אֲשֶׁר יְהוָה אֱלֹהֶיךָ נֹתֵן לָךְ. לֹא תִּֿרְצָח לֹא תִּֿנְאָף לֹא תִּֿגְנֹב לֹא תַעֲנֶה בְרֵעֲךָ עֵד שָׁקֶר. לֹא תַחְמֹד בֵּית רֵעֶךָ לֹא תַחְמֹד אֵשֶׁת רֵעֶךָ וְעַבְדּוֹ וַאֲמָתוֹ וְשׁוֹרוֹ וַחֲמֹרוֹ וְכֹל אֲשֶׁר לְרֵעֶךָ.
And God spoke all these words, saying: I am the Eternal your God, who brought you out of the land of Egypt, out of the house of bondage. You shall have no other gods before Me. You shall not make for yourself a graven image, nor any manner of likeness, of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth; you shall not bow down to them, nor serve them; for I the Eternal your God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate Me; and showing mercy to the thousandth generation of them that love Me and keep My commandments. You shalt not take the name of the Eternal your God in vain; for the Eternal will not hold him guiltless that takes His name in vain. Remember the sabbath day, to keep it holy. Six days shall you labour, and do all your work; but the seventh day is a sabbath to the Eternal thy God, in it you shall not do any manner of work, you,, nor your son, nor your daughter, nor your man-servant, nor your maid-servant, nor your cattle, nor the stranger that is within your gates; for in six days the Eternal made heaven and earth, the sea, and all that is in them, and rested on the seventh day; wherefore the Eternal blessed the Sabbath day, and hallowed it. Honor your father and your mother, that your days may be long upon the land which the Eternal your God gives you. You shall not murder. You shall not commit adultery. You shall not steal. You shall not bear false witness against your neighbor. You shall not covet your neighbor’s house; you shall not covet your neighbor’s wife, nor his man-servant, nor his maid-servant, nor his ox, nor his ass, nor any thing that is your neighbor’s.
According to this passage in the Mishnah (c. 200 CE), the 10 Commandments were part of the daily rite in the Temple:
אמר להם הממנה, ברכו ברכה אחת, והן ברכו.קראו עשרת הדברים, שמע, והיה אם שמוע, ויאמר, ברכו את העם שלש ברכות, אמת ויציב, ועבודה, וברכת כהנים.ובשבת מוסיפין ברכה אחת למשמר היוצא.
The superintendent said to them: “Say one blessing,” And they blessed. They recited the Ten Commandments; “Hear, [O Israel . . .]”, “And it shall come to pass, if you hearken . . .” (Deuteronomy 11:13–21), “And the Lord spoke . . .” (Numbers 15:37–41). They blessed the people with three blessings: “True and enduring” [the blessing that follows the Shema prayer in the morning prayer service], Avodah [the blessing that many Jews say as part of the Amidah prayer calling for Divine acceptance of the Temple service], And the Priestly Blessing (Numbers 6:24–26). And on the Sabbath, they added a blessing for the [priestly] watch that was leaving.
וקורין עשרת הדברות שמע והיה אם שמוע,ויאמר אמת ויציב ועבודה וברכת כהנים. א"ר,יהודה אמר שמואל אף בגבולין בקשו לקרות,כן אלא שכבר בטלום מפני תרעומת המינין
They recited the Ten Commandments, the Shema, the sections "And it shall come to pass if ye diligently hearken", and "And the Lord said", "True and firm", the Avodah, and the priestly benediction'. Rab Judah said in the name of Samuel: Outside the Temple also people wanted to do the same, but they were stopped on account of the insinuations of the Minim.
The "minim" or "sectarians" is a catch-all word for Sadducees, Samaritans, and early Christians. The followers of Paul held that only the 10 Commandments, not all of Mosaic law, was divine. Giving pride-of-place to the 10 commandments might signal agreement with this position.
The custom of standing for the public reading of the Ten Commandments, either during the weekly cycle or for Shavuot, seems to have arisen in the Middle Ages. It was hotly contested, as seen in this responsum from the Rambam (Moses Maimonides), written sometime between 1167-1200 :
From the Query:In one community they were accustomed to stand during the reading, until a wise rabbi came there and annulled that custom, instructing them to sit when it was read and preventing them from standing ... and he[the rabbi]included in his reply that anyone who wishes to stand when the Ten Commandments are read from the Torah scroll should be reproached, because to do so is the way of heretics who believe that the Ten Commandments have more importance than the rest of the Torah, and whenever the heretics believe differently from our Rabbis, of blessed memory, we must distance ourselves from them.

Maimonides' reply: That which the late rabbi instituted, to sit, is proper and his proofs are correct... and there is nothing to add to them. And it would be proper to do this in every place where they are accustomed to stand; one should prevent them from doing so because of the possible damage to belief, as some people may imagine that there are various levels of Torah and that only some parts are exalted, and that is a very bad thing... The claim of the opposing sage that in Bagdad and several other cities they did this[i.e. stood for the Ten Commandments], is by no means a proof. For if there are sick people, we do not make the healthy sick in order to render them all equal, rather we would try to cure each sick person... The Rabbis have already taught us that there is no difference between one who denies the Divine nature of the entire Torah and one who claims that a single verse 'Moses authored on his own'. There were among the heretics(minim) those who believed that only the Ten Commandments were given from Heaven and the remainder of the Torah was said by Moses himself, therefore the daily reading of the Ten Commandments was eliminated. And it is strictly forbidden to treat part of the Torah as if more exalted than another part.(Responsa of Maimonides, Blau Edition, Jerusalem, 1960, § 263).

“Leave Israel alone,” said Hillel, “for even if they are not prophets, they are still the children of prophets.” Ordinary Jews had a passion for the Ten Commandments. They were the distilled essence of Judaism. They were heard directly by the people from the mouth of God himself. They were the basis of the covenant they made with G‑d at Mount Sinai, calling on them to become a kingdom of priests and a holy nation...Jews kept searching for ways of recreating that scene, by standing when they listened to it from the Torah, and by saying it privately after the end of the morning prayers. Despite the fact that they knew their acts could be misconstrued by heretics, they were too attached to that great epiphany—the only time in history God spoke to an entire people—to treat it like any other passage in the Torah. The honor given to the Ten Commandments was the custom that refused to die.

Lord Rabbi Jonathan Sacks, Chief Rabbi of Great Britian and the British Commonwealth, "The Custom That Refused to Die."