It is common to talk about utility functions. For example in a universe with only two goods, we might assume each person (or group of people) carries a function $u(x,y)$ in their heads. When offered some baskets $B_j$ containing $x_j$ amount of good 1 and $y_j$ amount of good 2 the person will select the basket $B_k$ that maximises $u(x_k,y_k)$.

I wonder has anyone ever tried to find what there utility functions look like experimentally? Of course they might vary from person to person, and depend on the type of goods. But if the functions are a realistic model it should be possible in principle to see what they look like. Has there been any attempts to do so?

The closest I can find are regression attempts, where data is gathered and then fit to some class of functions. For example linear regression tries to draw the straight line (hyperplane) that best fits the data. This is not what I am looking for however, since regression assumes the type of function in advance. I'm looking for something as simple as offering a bunch of baskets, plotting the data, and comparing to a bunch of different types of functions.