  This paper endeavors to track the trajectories of individual horocycles on
\modsurf. It is far more common to study \emph{sets} of such trajectories,
seeking some asymptotic behavior using an averaging process (see section
\ref{previous}). Our work is only marginally related to these efforts.
  We begin by examining horocycles defined using the pencil of circles whose
common point (in the words of the Nielsen-Fenchel manuscript \cite{wF}) is
$\infty$. The orbits involved in this case are closed and long --- judged by
arc length between two points compared to the hyperbolic distance between them.
Using Ford circles of Farey sequences we find their lifts to the Standard
Fundamental Region (SFR) and find points of these lifts making given angles
with a horizontal. Next, we offer two algorithms, both involving continued
fractions, of locating points whose angle with the horizontal is near any
target angle and whose lifts are near any given point in the SFR.
  Next, we study the homotopy classes of horizontal horocycles as we descend to
the real axis. We find these are stable during descent between encounters of
the horizontal horocycle with elliptic fixed points. Such encounters change ---
complicate --- the homotopy classes. We give these explicitly down to height
$1/(2\sqrt{3})$.
  Finally we do an initial study of the open (infinite length) horocycle path
with unit euclidean radius anchored at $\phi -1$, where $\phi$ is the Golden
Mean. Enough information is adduced to suggest that this single doubly infinite
path is transitive.
