Really what you are asking about is "What should I expect when I am leaving/in the leave period of my employment at a company?"
You are leaving in a month for another opportunity. The reality of your situation is that involving employees(even contractors) into a project is an investment for a company. An investment in time, effort and training to get that person up and running.
This investment pays off for the company in several ways. Not only do they gather the fruit of the employee's labor but they are also 'investing' interesting and fun work in employees as a bid to keep those employees around. The talented superstars of a department are going to tend to be given more 'fun' projects not just because they can handle them but also as an incentive to stick with the company. As an aside, ideally companies would always try to match workers with projects that dovetail with their interests and strengths, but that is not always possible given business needs and market demands.
So, from this, investing in you doesn't gain them anything. You've already put in your notice, you are already halfway out the door. What is the benefit to the company of you getting involved in what sounds like the early stages of a project - you won't be around to finish it, any decisions you make won't effect you, and the rest of the team has to be able to handle it without you there(which makes spreading training to you especially dangerous.) Finally if this is a potential business product for the company your involvement could result in a competitor or other company getting information about the product before it is public. So with all of these risks what are the benefits here? Well you get to work on interesting projects but you have no risks. They have all the risks but no real benefits.
So since they aren't giving you interesting projects what should you do?
First, even though you are not getting interesting projects you should absolutely NOT work on external or personal projects on company time or with company equipment. In many employment contracts anything you work on on company time or with company equipment may be partially or completely owned by that company. Even beyond that, though, is the fact that this is stealing from your employer. Yes it sucks that you're not being giving cool things to do but that does not excuse the theft of your time.
Second, you should talk to your supervisor. You shouldn't ask for a part in the new projects, you should ask what you can do to make the transition easier for everyone involved. Is your documentation up to date? Could another person come in and find the resources and tools you use/have made to do your job? This is how you maintain business relationships like this beyond the end of a job. If you skate out without cleaning up after yourself(ie just do personal projects, prep work for your new job, and avoid grunt or unfun work) you will be burning bridges at your current company.