Expected nature of LISA's data; will it be more like a forest of static peaks, or a series of individual events? @RobJeffries' clear answer to Which things “LIGO can see things that LISA can't”, and vice-versa? explains that the Laser Interferometer Space Antenna or LISA will only be sensitive to gravitational waves with frequencies below about 1 Hz and so will be sensitive to much slower rotating objects than those that LIGO records.


  Lower frequency GWs can only be seen with LISA. This would include stellar binary systems with orbital periods longer than about 10 seconds, merging supermassive black holes and maybe GWs from the big bang


This can might include "event-like" signals, for example (possibly) a merger of two supermassive black holes that are so large in diameter that even during the merger their orbital frequency is below 1 Hz, but it can also include signals from the rotation of bodies not yet in contact but are slowly losing angular momentum though continuous GW radiation. While these would be much weaker, they would provide a very stable and very narrow peak in a spectrogram with an extremely slow drift, so you could accumulate data for years in order to bring them up out of the noise.

While LIGO's detected events have lasted of the order of 1 second or less, it sounds like it is possible that LISA could potentially record continuous GW radiation from a large number of pairs at the same time since the "lifetimes" might be years or millennia rather than seconds, provided that the sensitivity to these weaker signals is sufficient.

Question: What is the expected nature of LISA's data; will it be more like a forest of fairly static or slowly moving peaks in frequency space, or a series of individual events? Presumably this has been modeled based on some predictions of the number of different kinds of pairs at various separations expected to be found nearby.
LISA's data will be very different from LIGO's. It will typically "see" many sources at the same time. Most prominently:


Mergers of pairs of supermassive blackholes. These will be very much like a scaled version of the events that LIGO sees (probably more variation in the mass-ratio and ecccentricity though). They will be much louder (SNR>1000) than any event seen by LIGO though. They will (potentially) enter LISA's sensitivity band months before merger. Event rates are uncertain with estimates ranging from a few to hundreds per year. Depending on where this rate is in nature, we may get many overlapping signals.
Wide binaries These are binaries of compact objects (mostly white dwarfs) that are far from merger. They evolve very slowly, and will therefore mostly show up as continuous almost monochromatic signals. There will be thousands of such signals "visible" at any one time.
There has been some suggestion that LISA may be able to see the earlier stages of GW150914 like events months to years before merger. The current design limitations make this somewhat unlikely though.
Extreme mass ratio inspirals These are mergers of stellar mass compact object with a supermassive black hole. Due to the very small mass-ratio their evolution is very small, meaning that they will stat in the LISA sensitivity band years before merger. Their event rate is even more uncertain than that of supermassive BH mergers with estimates ranging from 1 to 1000s per year. In all likelihood these signal will overlap.


Consequently, LISA will be observing thousands of sources at any one time, which will be a significant challenge for the data analysis.