1819-1858: Singapore (and later as part of the Straits Settlements) was governed by the East India Company (EIC) on behalf of the British government.
1858-1867: the Straits Settlements came under direct British rule and were administered by the India Office in London.
The records of the EIC and India Office are preserved in the British Library and are called all together “India Office records” (reference code: IOR). NUS Libraries has acquired copies of some records pertaining to Singapore. These records are only available in microform format.
Main sources:
Straits Settlements Factory records [microform] : IOR/G34. Contents: IOR/G34/153-62, Singapore diary, 1827-1829 (9 reels) --IOR/G34/163-4, Singapore political & secret consultations, 1827-1828 (1 reel) --IOR/G/34/182A, Letters received from Prince of Wales Island, 4 Mar.-Jul. 1819, with enclosures (1 reel).
East India Company Records by Adam Matthews. Contents: Offers access to a unique collection of India Office Records from the British Library, London. Containing royal charters, correspondence, trading diaries, minutes of council meetings and reports of expeditions, among other document types, this resource charts the history of British trade and rule in the Indian subcontinent and beyond from 1600 to 1947.
Reproduction of manuscripts kept in the National Archives of India, New Delhi. Those records contains documents created or received by the Governor General of India and related to the Straits Settlements.
Reproduction of manuscripts preserved at the National Archives of Singapore. Each series is numbered from A to HH. Those records consist of documents created or received by the administrative entities located in the Straits Settlements (e.g. Governor of the Straits Settlements).
You can also consult Guide to the sources of history in Singapore, vol. 1 (a book published by the National Archives of Singapore which includes a description of the content of the different series).
Other sources:
The National Archives of Singapore (NAS) and the National Library Board (NLB) also have Straits Settlements Records. Search the NAS catalogue under "Straits Settlements Records" or the NLB catalogue under “Straits Settlements Records microform”