If there are no more constraints, then you can do it with arbitrarily low curvature with any reasonable class of splines.
If the points are say within a 10cm region, make huge loops 1km in diameter (or bigger if you want smaller curvature). If the spline construction is smooth, continuous, and invariant under similarity, then the curvature converges to 0.
If the curves are required to stay in a bounded region of the plane, then as the region gets smaller, not even arbitrary $C^2$ curves can thread through them with bounded curvature. Just imagine $3$ points at the corners of an equilateral triangle 1 micron on a side, and ask for the curve to be confined to a box 2 microns on a side. The curvature will be on the order of $\pi / $ micron.
Here are two copies of a set of four points threaded with Adobe Illustrator splines to illustrate the phenomenon. Note: I added extra knots in the big loops to make them look better, but this isn't necessary to construct examples. (The mathematical characterization of these splines is not relevant to the answer, and furthermore, I don't actually know):
alt text http://dl.dropbox.com/u/5390048/splines.jpg
The design considerations for splines are much more subtle than minimizing curvature.
However, I'd like to mention that the earlier meaning of splines had to do with thin splints of wood used in woodworking, e.g. boat building, to lay out curves for cutting. Unlike the usual mathematical splines, they have fixed length, and a reasonable
mathematical model is that they trace out curves that locally minimize total curvature subject to their constraints (lead weights called ducks because that's what they resembled).
It's easy to get examples of these traditional splines with multiple local minima: cut a strip of paper (good enough for this) and bring the ends closer without turning them. The strip pops to one side or the other, giving two local minima.
A good introductory lookup for 1) (and similar problems) is the book "Spectral Methods: Fundamentals in Single Domains" by Canuto, Hussaini, Quarteroni & Zang. Chapter 5, in particular. Equation (5.4.16) gives a bound for the $L^p$ norm approximation problem in terms of the L^p smoothness of $f$ and its derivatives:
$$
\inf_{q \in \mathbb{P}_n} \| f - q \| _{L^p} \leq C N^{-m} \left ( \sum^{m} _{k=\min(m,n+1)} \| f^{(k)} \|^p _{L^p} \right )^{\frac{1}{p}}
$$
According to the bibliographical notes section (p.291) a proof can be found in this paper.
Best Answer
Bounds in the univariate case, see e.g. here, were established by V.A. Markov in 1892.
S.N. Bernstein has given an extension of the result to the multivariate case in
Unfortunately I do not have access to that paper, but according to MR0023953, the following result is proved. Let $$P(x_{1},\ldots,x_{n})=\sum_{1\leq\alpha_{h}\leq d_{h}} A_{\alpha_{1},\ldots,\alpha_{n}}x_{1}^{\alpha_{1}}\ldots x_{n}^{\alpha_{n}},$$ such that $|P|\leq1$ on the cube $|x_{h}|\leq1$, $1\leq h\leq n$. Then $$|A_{\alpha_{1},\ldots,\alpha_{n}}|\leq\frac{\prod_{h=1}^{n}|B_{\alpha_{h}}^{(d_{h})}|}{\alpha_{1}!\ldots\alpha_{n}!},$$ where $B_{\alpha_{h}}^{(d_{h})}$ is the coefficient of $x^{\alpha_{h}}/\alpha_{h}!$ in the expansion of the Chebyshev polynomial $$T_{m}(x)=\cos(m\cos^{-1}(x))=\sum_{i=0}^{m}B_{i}^{(m)}x^{i}/i!,$$ and $m=d_{h}$ if $d_{h}-\alpha_{h}$ is even and $m=d_{h}-1$ if $d_{h}-\alpha_{h}$ is odd.
For the particular case of homogeneous polynomials $$P_{d}(x_{1},\ldots,x_{n})=\sum_{|\alpha|= d}c_{\alpha}x^{\alpha},$$ of total degree $d$, where $\alpha$ are multi-indices, such that $|P_{d}|\leq1$ on the ball $x_{1}^{2}+\cdots +x_{n}^{2}\leq1$, upper bounds on the coefficients are given in Theorem 2 of
namely, the coefficient of the monomial $x_{1}^{k_{1}}\ldots x_{n}^{k_{n}}$ cannot exceed in absolute value $$\frac{d!}{k_{1}!\ldots k_{n}!}.$$
For general (nonhomogeneous) polynomials of total degree $d$, there are also precise bounds on the coefficients $c_{\alpha}$ with $|\alpha|=d$ or $d-1$ (for the case of the ball or the cube).
In that connection, two interesting papers are
and the bibliography therein.