For print newspapers and microfilm facsimiles, start with SOLO where many will be listed and can be ordered in the normal way.
If you’re looking for a printed version of a pre 1800 newspaper and haven’t had any luck with SOLO, it is also worth consulting Milford and Sutherland:
Milford and Sutherland, A catalogue of English newspapers and periodicals in the Bodleian Library, 1622-1800. (Oxford, Oxford University Press, 1936)
This is an exhaustive A-Z of newspapers kept in the Bodleian Library. There are various copies around Oxford but it is best to use the one in the Lower Reading Room which is meticulously annotated – some of the annotations are vital if you trying to track down the print.
As you can see in the picture above, Milford and Sutherland provides detailed information about each publication. The text in square brackets are the shelf marks. However, do not attempt to order these newspapers yourself because some of the shelfmarks have changed. Find the item that you want in the catalogue and then take it to the staff on the Main Enquiry Desk, Lower Reading Room, Old Library. They will then assist you in placing your order.
A second source is the microfilm collection Early English newspapers. This comprises films of the majority of the Burney (British Library, available online for Oxford users) and Nichols (Bodleian) collections.
Some of these are on SOLO individually, mostly those with titles beginning A-L (although not all A-L). The others can be found by using the annotated version of Early English newspapers : bibliography and guide to the microfilm collection (1983 which is available at the Main Enquiry Desk, Lower Reading Room, Old Bodleian Library. Those marked o in pencil are on SOLO. The remainder need to be ordered from Early English newspapers. [Segment II] microform.
Details of Early English newspaper reels are often listed in the bibliography of newer catalogue records, even if pieces haven’t been added for them yet, or in the index of the guide. A microfilm reader printers are available in the Upper Reading Room and in the Weston Library.