How to construct a vector fields with isolated zeros? The Poincare-Hopf theorem tell us that the sum of the indices of a vector field at isolated zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold. But how to construct a vector fiedls with isolated zeros? 
 A: If one takes the differential of a Morse function, one gets
a differential form (a cotangent field) with isolated zeros.
If one has a Riemannian metric on the manifold one can
convert between covector fields and vector fields. So,
from a Riemannian metric and a Morse function you can write
down a vector field with isolated zeros.
A: Your question isn't very well defined.  A manifold on its own is not an object where constructions come by easily.  But there is a generic way to construct vector fields with isolated zeros.  Any vector field can be approximated by one with isolated zeros.  This is a consequence of Sard's theorem.  So start off with the zero vector field and choose any small random perturbation of that, and there you go. 
If you want a more constructive answer you'll have to assume a more constructive situation.  Like say if your manifold is triangulated, or has a handle decomposition, or a morse function.  
Chapman describes the Morse situation so I'll give the triangulation situation. 
The vector field has these properties: 
There is a critical point at the barycentre of every cell in the triangulation.  The vertices are repellors.  The barycentres of the top-dimensional simplices are the attractors.  A 1-simplex is a (1,n-1)-index critical point -- meaning there's two orbits approaching (along the 1-simplex) and an n-2-dimensional family of reverse orbits attracting.  Etc.  A j-simplex barycentre has a j-1-dimensional family of attracting orbits, and an n-j-1-dimensional family of reverse orbits attracting. 
That isn't quite explicit as one needs an explicit smoothing of the triangulation to put this all together.  But it gives you the idea. 
A: Just use the transversally theorem, an application of Sard's theorem: the generic vector field intersects the zero-section of the tangent bundle transverse, therefore the zeros are isolated.
