I've listed out a couple of instances where Leibniz's notation is abused, sometimes with nasty consequences. What are some other situations where Leibniz's notation is abused?
Chain Rule
Let $u = g(x)$ and $y = f(u)$, then: $$(f(g(x)))' = \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$
Even with differential forms, it is still a bit awkward. Let $u(x) = g(x)$ and $y(x) = f(u)$, then: $$\frac{dy}{dx} = \frac{ \frac{df}{du} du}{dx} = \frac{ \frac{df}{du} \frac{du}{dx} dx}{dx} = \frac{df}{du} \frac{du}{dx}$$
There's plenty of good answers which discuss its problems and alternatives. But just to add on to it, here's a common example of its abuse:
$$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx} = \frac{dy}{dt} \frac{1}{ \frac{dx}{dt} } = \frac{y'(t)}{x'(t)}$$
This method is commonly shown in multivariable calculus courses when "deriving" curvature. Often called the dummy variable approach to "deriving" curvature. It's featured in LibreTexts and plenty of educational YouTube videos "deriving" curvature. An actual derivation using Lagrange's notation is significantly more enlightening.
Multivariable Chain Rule for implicit functions
In Leibniz's notation, we can obtain the partial derivative for the implicit function $F(x, y, z) = 0$ as:
$$\frac{\partial z}{\partial x} = - \frac{\partial F / \partial x}{\partial F / \partial z}$$
Which can cause the contradiction: \begin{aligned} \frac{\partial z}{\partial x} = - \frac{\partial F / \partial x}{\partial F / \partial z} \\ \frac{\partial z}{\partial x} = - \frac{\partial F}{\partial x} \frac{\partial z}{\partial F} \\ \frac{\partial z}{\partial x} = - \frac{\partial z}{\partial x} \end{aligned}
In Lagrange's notation, this is a non problem: $(z)'_x = - \frac{F'_x}{F'_z}$. This kind of contradiction has been discussed in, other answers. The contradiction appears due to a misunderstanding/abuse of Leibniz notation which is very easy to make.
Integration by substitution
Integration by substitution, or u-substitution, often includes a similar process. As an example:
$$\int (x+1)^2\ dx$$
Let $u=x+1$, then $\frac{du}{dx} = 1 \implies dx = \frac{du}{1}$
$$\int (x+1)^2\ dx = \int (u)^2\ \frac{du}{1}$$
Which is a helpful mnemonic, yet the same rule can be derived quite nicely from the chain rule, as it's simply dividing out the $g'(x)$ produced by the chain rule. Again, this has been discussed in other questions.
Separation of variables
This one is named after the abuse of notation. Unfortunately, it rather obscures the fact that it's simply integration by substitution: $$\frac{dy}{dx} = g(x) \cdot h(y)$$ $$dy = g(x) \cdot h(y) \cdot dx$$ $$\frac{1}{h(y)} dy = g(x) \ dx$$ $$\int \frac{1}{h(y)} dy = \int g(x) \ dx$$
This whole splitting step does almost seem mystical, there are plenty of great answers and alternatives to this method. It is unfortunate however that the name "Separation of variable" has stuck, which goes to show how prevalent the abuse of notation is.
Green's theorem
$$\int_C Pdx + Qdy = \int_C (Pdx + Qdy) \frac{dt}{dt} = \int_C \left(P\frac{dx}{dt} + Q\frac{dy}{dt}\right) dt$$
While a bit unfair, since it is simply a differential form, the reasoning offered to students can only be characterised as an abuse of notation. Personally, I feel that hinting/suggesting differential forms is unnecessary and unhelpful in most multivariable calculus course.
