The claim is that socialism would bring about a fairer society than our present one. Society could indeed be made fairer, but socialism is not the way to do it, and never has been. Whenever socialism has been tried it has involved compulsion, as it attempts to make people behave in ways they would not freely choose to do.
Socialism has always led to economic deprivation and shortages. Markets allocate resources according to demand and anticipated demand, whereas socialism attempts to achieve this by planning, planning by the few instead of by the many. The rulers make their five-year plans, but the attempt to anticipate what people will want and need is always inferior in practice to the activities of entrepreneurs and businesses as they attempt to profit by meeting future demand. The spontaneous market economy is far superior to the planned socialist one. The market economies grew wealthy; the socialist ones stayed poor.
Moreover socialism has in the past led to murder, sometimes of political dissidents, and sometimes mass murder on a horrendous scale. The consensus among analysts is that socialist regimes last century murdered 100 million of their own people. Obviously the Soviet Union, Communist China and Cambodia were among the leaders, but the other socialist countries contributed to the total.
Most of those who advocate socialism tell us that it has never been tried, and that the socialist countries never implemented 'true' socialism, which is a theoretical concept yet to be tried in practice. Yet the socialist countries called themselves socialist; they acted in the name of socialism; they thought they were practising socialism; and many modern socialists are apologists for what they did. If it's called a duck, looks like a duck, has feathers and webbed feet and a beak like a duck and it quacks like a duck, it's a duck.