It's always hard to say what benefits someone can or can't get on the limited information you have given, however I can tell you the basic rules.
You can only get Income Support if you are within a "prescribed category of person" this is effectively either:
a lone parent with children under 16
sick/disabled and incapable of work
caring for so,ebody who is sick
within 10 years of your 60th Birthday
If you are working over 16 hours a week or have a partner living with you who is working over 24 hours a week you can't receive it, even if you fit in the above categories.
Unemployment was replaced by Jobseekers Allowance in 1995, it is paid in two amounts, contribution based and income based. The contribution based element requires you to have paid enough national insurance for the past two fuill tax years and will only be payable for 6 months. If you leave your job voluntarily or are sacked, you will not be entitled.
The income based element is means tested the same way as Income Support is - savings over £8000 will disentitle you as will any income, or partner working. Again if you leave your job you will not receive anything for 6 months.
In both cases you have to be signing on as available for employment and you have to accept any job that is offered - students are not entitled since they are not considered to be available for employment.
hope that helps.