Should I put my last job down as a reference

I lost my job on Monday. I am now trying to figure out if I should put them down on my resume or job applications. If I do I would have about i 9 months of experience, if I don't I would only have 6 months. I asked my boss if I could use her as reference, because she said I have a lot of potential, but she said she can't due to corporate. I am going to use some CEUs to make up for the mistakes I made so I won't mess up again, but I just don't know if I should put them down or what.
Put it down because you want to be honest. If the new place calls to check on you HR will only tell them your dates of employment and nothing else. You can call HR from where you were fired and ask what info they will give to a potential employer. Then you could say the place was not a good fit for you because it wasn't from what I have read in your other post.
Thank you for your help St._Claire! I appreciate the advice you have given me.
You should include your last employer. What will you say to explain the gap otherwise? Be prepared to discuss what you've learned from your mistakes and how you will improve.