What is an example application of a quasilinear utility function? I am told a quasilinear utility function is a function like $$U(x,y) = \sqrt{x}+y$$

My Question:

Can someone provide a real world example of a quasilinear utility function?
Quasilinear utility functions are useful in much of the demand estimation literature, particularly in discrete choice. For instance, check out Berry 1994,Berry Levinsohn Pakes 1995 and the many applications in Nevo's papers on demand estimation (here's a "practicioner's guide"). Ken Train's book on it is available for free here!

To summarize, they can lead to indirect utility of the form $$u_{ijt}=\alpha_i\underbrace{(y_i-p_i)}_\text{real income}+\underbrace{X{jt}\beta_i}_\text{observed product characteristics$*\beta_i$}+\underbrace{\xi_{jt}}_\text{unobserved product characteristics}+\underbrace{\epsilon_{ijt}}_\text{mean zero stochastic term}$$
where $i$ represents individuals $i=1,\dots,I_t$ in each of the $t=1,\dots,T$ markets selling $j=1,\dots,J$ products. Here, $\alpha_i$ represents the marginal utility of income and $\beta_i$ represents the marginal utility from the product characteristics observed in $X_{jt}$.

Suppose that we restrict the heterogeneity across consumers to only enter through the stochastic term $\epsilon_{ijt}$. Then both the individual specific parameters $(\alpha_i,\beta_i)$ must be equal to $(\alpha,\beta)$ and the market share of each good can be represented by
$$s_{jt}=\frac{exp(X_{jt}\beta-\alpha p_{jt}+\epsilon_{jt})}{1+\Sigma_{k=1}^{J}exp(X_{kt}\beta-\alpha p_{kt}+\xi_{kt})}$$

The equation for the market share is a function of variables that only vary at the product-market level, so you only need information on prices and quantities (and product characteristics) for a crude estimation. However it gives some quirky results. Particularly regarding the elasticity of market share wrt own price and cross price (market share) elasticities, and consumer subsitution patterns. Things can get more robust the more you read on demand estimation, you can introduce consumer characteristics and get results that have more desirable substitution properties. 

There are also many criticisms of the conclusions that this type of modeling can draw, but I'll leave that up to you to imagine should you want to read on the topic.