It's very easy to confuse "Social Security benefits" (which always need to be reported on the 1040 and MAY be taxable) with "Supplemental Security Income (SSI)," which is never taxable and should NOT be reported on any taxpayer's return.
The Social Security Administration sends out both kinds of payments, so the confusion is understandable. However, there is an important conceptual difference between the two programs.
Social Security benefits are an example of categorical transfer payments that do NOT have a means-test. That is, you can collect Social Security even if you are extremely wealthy, as long as you fit into the eligible category (based on age, permanent disability, or survivorship status.) Under current law, even someone as rich as Bill Gates will be eligible for Social Security benefits when he reaches retirement age, regardless of how many billions of dollars he has. That's because Social Security is a "social insurance program" in which benefits are (somewhat loosely) tied to payroll taxes that a worker pays during his working years. When Bill Gates collects Social Security, most of his benefits will be taxable. A retired person who has very little income other than Social Security benefits will probably not have to pay tax on any of it.
Supplemental Security benefits are an example of "means-tested" benefits. They go only to elderly or disabled persons whose incomes fall below specified levels. Since SSI recipients can only qualify if their income is extremely low, it does not make sense to tax that income.