One of the offences took place at the Heron store in Eastgate on Tuesday (May 15) at approximately 2pm.
The man is described as white, in his 50s, of large build, with distinctive tattoos on his arms, a moustache, and long brown hair.
If you know anything about this incident, or knows who the man in the picture is, call the police on 101 - quoting incident number 218 of May 15.
Police would also like to speak to the two women pictured in connection with alleged shoplifting at the Co-op in Newbridge Hill on Saturday May 5.
If you have any information, or recognise either of the women, call police on 101 - quoting incident 214 of May 5.
For either incident, you can alternatively contact Lincolnshire Police via email at [email protected] (remembering to include the incident number and the date in the subject box), or via the independent charity Crimestoppers anonymously on 0800 555111 or online at crimestoppers-uk.org