A major red flag is gaps and short employment. Make sure you explain any gaps in your resume on your cover letter. Only put a job that you had for a month on your resume is it is your only relevent experience or it is really necessary to you getting the job.
Mention it only if you can explain why you left within a short time. Of course the reason should not create a negative impression. For eg. If you say you left for more remuneration it may sound like you are only after money. If you changed due to family exigencies that may sound genuine.
I believe that all jobs you have had should be included on your resume - as long as you state for how long you had the job. As long as you are honest, and you don't mislead anyone, then I believe that this can be included.