Epigraphic Old Arabic is the name given to those pre-Islamic texts in the Arabic language that — unlike the pre-Islamic Arabic poetry and the Ayyām al-ʿArab — have survived independently, rather than being transmitted through the scholars of the Islamic period. The term 'epigraphic' is used because most of the texts that have survived independently are inscriptions.

Before the fifth century AD, Arabic was hardly ever written down and so there was no dedicated 'Arabic script'. Speakers of the various forms of Arabic had to use a different language for writing, such as Nabataean Aramaic or one of the Ancient North Arabian dialects or Greek or, possibly, Sabaic. So, on the rare occasions when someone wanted to write in their spoken language, Arabic, they would generally use the script of their habitual written language, such as Nabataean, Greek, an Ancient North Arabian script, etc.

We have only a handful of texts which are clearly in epigraphic Old Arabic of which the following are the clearest:

1. Two lines of rhetorical Arabic written in the Nabataean Aramaic script and embedded within a Nabataean graffito found near the Nabataean town of ʿOboda in the Negev. The date is uncertain. See Hackl, Janni & Schneider 2003: 396–402, and the bibliography there.
2. The five-line epitaph of Mrʾlqys son of ʿmrw king of all ʿrb (a large area between the Tigris and Euphrates) dated to AD 328, which was found over the door of his mausoleum at al-Namārah in the desert of southern Syria. See, for instance, Bordreuil et. al 1997 and Macdonald (in press).
3. There are large numbers of personal and place names of Arabic etymology in Greek inscriptions and documents dating from the last two centuries BC to the seventh century AD. In addition, there is a fragmentary parchment with the Septuagint text (in Greek) of Psalm 77 (78 in today's numbering) in one column and an Arabic gloss written in a parallel column. The dating of this is disputed but it seems likely that it is pre-Islamic (see Al-Jallad forthcoming). On Greek used to write Arabic see Al Jallad 2014a.
4. At the site of the ancient city of Qaryat al-Fāw, on the north-west edge of the Empty Quarter in southern Saudi Arabia, a tomb inscription was found which although in the Sabaic script has always been thought to be in the Old Arabic language. However, this has now been questioned by A.M. Al-Jallad 2014b.
5. A tomb inscription of uncertain date, in the Dadanitic script (Jaussen and Savignac 1909–1914, ii: Lihyanite 384), which contains the Arabic feminine relative pronoun allatī (Müller 1982: 32–33). For the most recent study see Macdonald (forthcoming).
6. A graffito in the Nabataeo-Arabic transitional script (on which see Nehmé 2014) in which an Arabic sentence snt 350 ʾdḫlw ʿmrw ʾl-mlk 'the year 350 [when] they introduced ʿAmr the king' (see Nehmé 2010: 76–77) is preserved among 'fossilized' Aramaic expressions (see Macdonald 2010b: 20). The date is equivalent to AD 455–456.
7. A list of names following a damaged invocation in what is recognizably the Arabic script. It is carved on the lintel of a martyrion at Zebed, northern Syria, and dated to AD 512. See most recently Macdonald (in press).
8. A graffito at Jabal Says in southern Syria dated AD 528, which is also clearly in what we think of as the 'Arabic script'. See most recently Macdonald (in press) and also Larcher 2010 and Macdonald 2010c.
9. An Arabic-Greek bilingual inscription on the lintel of a martyrion at Ḥarrān in southern Syria, also in the Arabic script. It is dated to AD 568. See most recently Macdonald (in press).

There are a number of other possible examples of Old Arabic but these are disputed. See Macdonald 2008.

References

Bordreuil, P., Desreumaux, A., Robin, C., and Teixidor, J. 1997. 205. Linteau inscrit : AO 4083. Pages 265-269 in Y. Calvet and C.J. Robin, Arabie heureuse Arabie déserte. Les antiquités arabiques du Musée du Louvre. (Notes et documents des musées de France, 31). Paris: Éditions de la Réunion des musées nationaux.

Hackl, U., Jenni, H. & Schneider, C. 2003. Quellen zur Geschichte der Nabatäer. Textsammlung mit Übersetzung und Kommentar. (Novum Testamentum et Orbis Antiquus, 51). Freiburg Schweiz: Universitätsverlag / Göttingen: Vandenhoeck & Ruprecht.

Al-Jallad, A.M. 2014a. Graeco-Arabica I: The Southern Levant. F. Briquel-Chatonnet, M. Debié et L. Nehmé (éds), Le contexte de naissance de l'écriture arabe. Écrit et écritures araméennes et arabes au 1er millénaire après J.-C. Actes du colloque international du projet ANR Syrab (Orientalia Lovaniensa Analecta). Louvain: Peeters.

Al-Jallad, A.M. 2014b. On the Genetic Background of the ʿIgl bn Hfʿm Grave Inscription at Qaryat al-Faw, Bulletin of the School of Oriental and African Studies 77.

Al-Jallad, A.M. Forthcoming. Old Arabic: an evidence-based history.

Larcher, P. 2010. In search of a standard: dialect variation and New Arabic features in the oldest Arabic written documents. Pages 103–112 in Macdonald, M.C.A. 2010a.

Macdonald, M.C.A. 2008. Old Arabic (Epigraphic). Pages 464–477 in K. Versteeg (ed.), Encyclopedia of Arabic Language and Linguistics, volume 3. Leiden: Brill.

Macdonald, M.C.A. (ed.) 2010a. The development of Arabic as a written language. (Supplement to the Proceedings of the Seminar for Arabian Studies volume 40). Oxford: Archaeopress.

Macdonald, M.C.A. 2010b. Ancient Arabia and the written word. Pages 5-28 in Macdonald 2010a.

Macdonald, M.C.A. 2010c. The Old Arabic graffito at Jabal Usays: a new reading of line 1. Pages 141–143 in Macdonald 2010a.

Macdonald, M.C.A. (in press). The emergence of Arabic as a written language. In G. Fisher (ed.), Arabs and Empires before Islam. Oxford: Oxford University Press.

Müller, W.W. 1982. Das Altarabische der Inschriften aus vorislamischer Zeit. Pages 30-36 in W. Fischer (ed.), Grundriß der Arabischen Philologie. Band I: Sprachwissenschaft. Wiesbaden: Reichert.

Nehmé, L. 2010. A glimpse of the development of the Nabataean script into Arabic based on old and new epigraphic material. Pages 47-88 in Macdonald 2010.

Nehmé, L. 2014. Can one speak of Arabic script in Arabia in the 5th century AD ? in F. Briquel-Chatonnet, M. Debié et L. Nehmé (éds), Le contexte de naissance de l'écriture arabe. Écrit et écritures araméennes et arabes au 1er millénaire après J.-C. Actes du colloque international du projet ANR Syrab (Orientalia Lovaniensa Analecta). Louvain: Peeters.