Although pairs of MOLS of order 10 are known ("Euler spoilers"), the existence of three mutually orthogonal latin squares of order 10 is an open problem.
You may find some details of computer-based searches for these in Erin Delise's M.Sc. thesis (2005). Note the first two bibliographic entries, for Bose, Chakravarti, and Knuth (1960-61).
There are "tricks" to reducing the general search space, but it remains of a formidable size despite its numerous symmetries. The first row of all three such matrices may be assumed to be $(1,2,3,\ldots,10)$, and the number of individual latin squares of order 10 with this fixed first row is 2750892211809148994633229926400. We may further restrict the first column of the first (of the three) latin squares to be $(1,2,3,\ldots,10)^T$, saving a factor of $9!$ in size of search space.
Once an intial pair of MOLS are fixed, it becomes tractable to make an exhaustive computer search for the third MOLS of order 10. Many attempts along this line have failed, although an "almost orthogonal" triple was reported by Franklin (1983); see Mohan, Lee, and Pokhrel (2006) for some references to the literature.
A set of 9 mutually orthogonal latin squares of order 10 would amount to the existence of a finite projective plane of order 10, but Lam (1991) reported the results of an extensive computer search proved this impossible.
Best Answer
How comfortable are you with finite field arithmetic? For latin squares of size a prime power $q$ there is a very nice construction for producing $q-1$ mutually orthogonal latin squares. Here it is for $q = 4$:
First identify the $4 \times 4$ grids with $\mathbb{F}_4^2$ in the obvious way. For each nonzero element $a$ of $\mathbb{F}_4$ we construct a latin square by labeling the points of $\mathbb{F}_4^2$ as follows: For a point $p \in \mathbb{F}_4^2$ take the line through $p$ of slope $a$ and label $p$ by where this line intersects the $x$-axis.
Since two points in $\mathbb{F}_4^2$ define a unique line it follows that these are mutually orthogonal latin squares.