The defining features of bell shaped functions:

A bell-shaped function or simply 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve. These functions are typically continuous or smooth, asymptotically approach zero for large negative/positive $x$, and have a single, unimodal maximum at small $x$.

Useful distributions in physics tend to have those traits because: continuous/smooth functions are useful for calculus rules to be followed, asymptotic approach to zero leads to finite values when integrating over $\mathbb R$ and also small/no chance of having object at those limits.

Useful distributions in physics tend to have the following traits:

• continuous/smooth function
• asymptotic approach zero for large $x$ and either very small $x$ (i.e., 0) or negative infinity
• have a single peak

which are pretty much the defining features of bell shaped functions:

A bell-shaped function or simply 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve. These functions are typically continuous or smooth, asymptotically approach zero for large negative/positive $x$, and have a single, unimodal maximum at small $x$.

There are, of course, useful distributions in physics that do not follow all of these traits (and therefore are not bell-shaped). For instance power-law distributions (used in stellar initial mass function and cosmic ray fluxes), this type of distribution still is continuous and single-peaked, but does not asymptotically approach 0 at either end. In this case, when one needs to integrate over the distribution, one would use the physical bounds for the upper and lower limits (e.g., 0.08$M_\odot$ and ~150$M_\odot$ for the initial mass function, cf. this SE post of mine), rather than $(0,\,+\infty)$ or $(-\infty,\,+\infty)$