If you have no National Insurance contributions (or PRSI, which is the equivalent here), you can only apply for a means-tested scheme, and entitlement will depend on your wife's income.
If you are available for work, you can apply for Jobseeker's Allowance.
If you are unable to work due to illness or disability (long-term), you can apply for Disability Allowance.
If you have no income and are not long-term ill or available for work, you can apply for Supplementary Welfare Allowance, only if your wife works less than 30 hrs per week - if she works 30 hrs or more per week you aren't eligible regardless of her income.
All of the above schemes are subject to habitual residence.
There are 5 factors taken into consideration when the assessing habitual residence:
Factor 1 - Length and continuity of residence in Ireland or in any other particular country
Factor 2 - Length and purpose of any absence from Ireland
Factor 3 - Nature and pattern of employment
Factor 4 - Applicant's main centre of interest
Factor 5 - Future intention of applicant concerned as they appear from all the circumstances.