Socialism isn't incompatible with democracy or even a market economy. Socialism is not a monolithic ideology, and is broader than Leninism or Maoism.
There are varying degrees of certain aspects within socialism, for instance, the current governance in Laos or Vietnam isn't the same or quite as authoritarian as a North Korea or a 1980's Cuba.
However there are a few hard attributes that would have to be adhered to in order for a nation to be socialist by the true definition of the word...one being ownership over the means of production and large amounts of government/public ownership over industry sectors.
Having a market economy with some more generous social safety nets, as the Danish PM stated, doesn't equate to socialism.
Another key aspect (though not a hard fast rule, but a trait that's replete in socialism) is that you can't typically vote your way out of it. Many of the truly socialist states have the hallmark of single-party rule, or one dominant party, with a few faux parties tossed in the mix (that have no real power or policy influence) to give the appearance of legitimizing elections, but with very strict rules pertaining to which parties are allowed to run candidates once the socialist-leaning party takes a majority. (Sort of the 'pulling up the ladder behind you' mentality)
For instance, if we look at the current Leninist states that still exist (N Korea, Laos, Vietnam), they all have single-party rule.
If we look at some of the non-Leninist states...
Bangladesh, Nepal, Sri Lanka, Nicaragua
They carry the latter trait I mentioned, which is that even though it's "multi-party", it's "dominant party rule", meaning that the socialist parties have the vast majority of seats, and put laws in place drastically limiting the number of people from other parties who are allowed to run.
For instance, in Bangladesh, their socialist party holds 298 seats in their legislative branch, the two center-right parties combined have 33 seats, and the controlling party has put a rule in place stating the conditions (that favor the controlling party) for being able to run for additional seats.