Newspapers
Where to Find Them and What You Might Find

Local Newspapers
The British Library at Colindale, London has the following, mainly original newspapers.  In my experience some years are in poor condition and cannot be viewed:

Malton & Norton Gazette and General Advertiser: June 1855 to December 1857
Malton Gazette: December 1857 to December 1884
North-Eastern Advertiser, Malton Gazette and Ryedale Chronicle: January 1885 to December 1887.
Malton Gazette January 1888 to September 1905.
Malton Mail: October 1898 to January 1899 (this newspaper then amalgamated with the Yorkshire Observer).
Malton Messenger: July 1855 to September 1939.

Malton Library has the following:

Malton Messenger April 1854 to September 1939 mainly on microfilm
Malton Gazette from 1819, some on microfilm, some original copies and I'm afraid some unusable due to poor condition.  There's about 10 years missing, from 1954.

The York Herald has coverage of some Malton events.  York Reference Library has an almost complete run of this from 1790 through to 1889 and then as the Yorkshire Herald through to 1954.  See also the library's newspaper index.  Malton Library also has the Yorkshire Gazette from 1819 to 1854.

A full listing of the whereabouts of old Yorkshire newspapers can be found here.