  Metric Temporal Logic, $\mtlfull$ is amongst the most studied real-time
logics. It exhibits considerable diversity in expressiveness and decidability
properties based on the permitted set of modalities and the nature of time
interval constraints $I$. \oomit{The classical results of Alur and Henzinger
showed that $\mtlfull$ is undecidable where as $\mitl$ which uses only
non-singular intervals $NS$ is decidable. In a surprizing result, Ouaknine and
Worrell showed that the satisfiability of $\mtl$ is decidable over finite
pointwise models, albeit with NPR decision complexity, whereas it remains
undecidable for infinite pointwise models or for continuous time.} In this
paper, we sharpen the decidability results by showing that the satisfiability
of $\mtlsns$ (where $NS$ denotes non-singular intervals) is also decidable over
finite pointwise strictly monotonic time. We give a satisfiability preserving
reduction from the logic $\mtlsns$ to decidable logic $\mtl$ of Ouaknine and
Worrell using the technique of temporal projections. We also investigate the
decidability of unary fragment $\mtlfullunary$ (a question posed by A.
Rabinovich) and show that $\mtlfut$ over continuous time as well as
$\mtlfullunary$ over finite pointwise time are both undecidable. Moreover,
$\mathsf{MTL}^{pw}[\fut_I]$ over finite pointwise models already has NPR lower
bound for satisfiability checking. We also compare the expressive powers of
some of these fragments using the technique of EF games for $\mathsf{MTL}$.
