The siege of Herculaneum (probably to 11 June 89 BC) is one of the more obscure incidents of the Social War, and saw a Roman army recapture the town after it fell to the Samnites in 90 BC.
During 90 BC the Samnite leader Gaius Papius Mutilus captured Nola, east of Naples, and then moved south to take Stabiae, Surrentum and Salernum and plunder the area around Nuceria, all in the area to the south of Pompeii, between the Gulf of Naples and the Gulf of Salerno. He then attempted to capture Acerrae, between Naples and Nola. Appian reports at least two Roman attempts to lift the siege, but doesn't actually say how it ended.
Appian doesn't mention the fate of Herculaneum or Pompeii, but both probably fell to Papius Mutilus during this phase of the war.
Appian also fails to mention Herculaneum or Pompeii directly in his account of Sulla's campaign in Campania in 89 BC, but we can infer that his original aim in this campaign was probably to recapture both places and the other nearby towns lost in 90 BC. The account begins with Sulla encamped in the Pompeian hills, before a Samnite relief army under Lucius Cluentius camped nearby. After an early setback Sulla defeated this army, and pursued it north to Nola, where he killed Cluentius and 20,000 of his men.
Velleius Paterculus says that Titus Didius and Velleius's own ancestor Minatius Magius captured Herculaneum (alongside Velleius's own ancestor Minatius Magius of Aeclanum), and that Magius then went on to attack Pompeii alongside Sulla. Minatius Magius was from the Samnite town of Aeclanum, and had raised a loyalist legion amongst the Hirpini. This reminds us that not all of Rome's allies sided with the rebels - the Hirpini are listed amongst the rebel forces, and Sulla went on to capture Aeclanum and subdue the Hirpini after defeating the Samnite relief army.
In Ovid's Fasti (On the Roman Calendar) the death of Titus Didius is placed on 11 June 89 BC. Although neither of our sources explicitly state that Didius died while storming the town, this is now the general assumption
The exact sequence of events is thus unclear, but it seems unlikely that Sulla would have defeated the relief army and then left the sieges of Herculaneum and Pompeii incomplete while he campaigned elsewhere. The most likely sequence is thus that Sulla arrived in the area before 29 April 89, when Stabiae fell to him. He besieged Herculaneum and then Pompeii. Cluentius's relief army arrived in the area while at least one of these sieges was still under way, although it could have been before or after the fall of Herculaneum. Sulla defeated Cluentius and pursued him to Nola, then returned south to complete whichever of the sieges was still underway. After then he moved into the Apennines, starting with Aeclanum, twenty five miles to the east of Nola.