Notation: $\nabla \cdot$, div, $\nabla$, grad, ...?
I, for myself, prefer the spelled-out variants, at least when doing analysis: $$\mathop{\mathrm{grad}} f, \quad \mathop{\mathrm{div}} \mathbf F, \quad \mathop{\mathrm{rot}} \mathbf F $$ The variant $\nabla f$ for the gradient is ok, but I like consistency, and using the same "style" underlines the connection between the three operations (more below). Sometimes, however, especially when doing physics, I need to revert to the "physicist $\nabla$ notation" because some formulae look a lot more familiar that way, e.g. $$\nabla \cdot \mathbf E = \frac{\rho}{\varepsilon_0} $$ For the laplacian, instead, I very much favor the notation $\Delta f$, since I find it easy to confuse $\nabla^2 f$ with $\nabla f$ or the bilaplacian $\Delta^2 f$ at first glance. Plus the big triangle is something specific to the laplacian, which makes it recognizable to me – it screams "heat equation". Same for the dalembertian $\square f$ (which I write in the mathematician's way, without the $^2$). $$\Delta u = \beta \frac{\partial u}{\partial t}, \qquad \square u = \frac 1 {c^2} \frac{\partial^2u}{\partial t^2} - \Delta u = 0 $$ It all changes when I need to do tensors on manifolds. All the first-order derivative operators are instances of the exterior derivative on forms (of different degree), so it is best to keep it simple: $$F = F_{\mu\nu} \mathop{}\mathrm{d}x^{\mu}\wedge \mathop{}\mathrm{d} x^\nu = B + E \wedge \mathop{}\mathrm d x^0, \qquad \mathrm d F = 0 $$ If you want to be acting on vector fields or scalar fields on three-dimensional manifolds specifically, though, you may "redefine" the differential operations with the Hodge star, the external derivative, and the musical isomorphisms: $$\mathop{\mathrm{grad}} f = (\mathrm d f)^\sharp, \quad \mathop{\mathrm{div}} X = \star \mathrm d{\star}(X^\flat), \quad \mathop{\mathrm{rot}} X= \left(\star \mathrm d (X^\flat) \right)^\sharp$$ Heck, you may even redefine the laplacian as $\mathop{\mathrm{grad}}(\mathop{\mathrm{div}})$: $$\Delta f = \star \mathrm d {\star} \mathrm d f$$ Also, in the context of differential geometry I prefer to denote the standard real-analytic partial derivatives as $\partial_k$, leaving the $\partial / \partial x^k$ to be abstract vectors on a manifold with respect to a chart $x$.
Bonus. Here is a diagrammatic depiction of the action of the first order differential operators on $p$-forms over a $3$-dimensional manifold $M$:
Here $\Omega^p$ denotes the space of $p$-forms $\Omega^p(M)$, while $\Gamma = \Gamma(TM)$ is the space of smooth sections of the tangent bundle, i.e. vector fields. $0$ is just the space $\Omega^4= \Omega^5 = \dots$, which is the trivial vector space.