There are a number of firmwares available for the Topfield TF5800, TF5800t and TF5810 PVRs. These may be categorised as Official, Semi-Official and Unofficial, with the latter two incorporating various patches. Included below are explanations of the main differences and some Recommendations.
Please note: links and information were correct at the time of writing, but you should be able to find the most up-to-date information by visiting the referenced main site and navigating to the specific product/download areas.
Official firmwares are available from the Topfield.co.kr site. None of them are recommended.
Semi-official firmwares are available from the Topfield.co.uk site — these contain some essential patches on the official firmwares.
Unofficial firmwares are also available from the Toppy.org.uk site. The main, public list of firmwares contains Official (unpatched) firmwares including some beta releases not available elsewhere, but they are now a little out of date. That list is also a bit misleading since better firmwares are available to registered members in the member area (it's not entirely obvious, but you will need to register with the forums part to gain access to that area.
TF5810 owners have little choice but to use a 5.15.X firmware, which is the same as the corresponding (and not recommended) TF5800 5.14.X firmware but with changes for support of the TF5810-specific hardware (the HDMI up-scaler and the better Front Panel). The main problems are discussed here. The current best starting point is 5.15.09+ plus any extra patches described in that thread.
For some reason, Topfield decided to give this face-lift version a new SysID (i.e. it will show "C458" on the Front Panel when booting, rather than the "C456" of the normal TF5800), even though the hardware is identical and the unit is otherwise completely compatible with the normal TF5800 (in fact, some normal TF5800 models are also believed to have shipped with this 458 SysID).
Any TF5800 firmware will work on the TF5800t, but the different SysID prevents direct use and may also confuse some TAPs. The latter is easily remedied by the SysID 456 [Sy] patch, but the former requires using VegaPack to alter the firmware (.tfd) file to SysID 458 or using HDFW to force on the firmware (including aXeL [Xl]).
The Official firmwares are (currently) way out of date and not recommended at all. And, in general, the 5.14.X firmwares are not recommended because Topfield's implementation of the Freeview+ system (including Series Links and Accurate Recording) is somewhat lacking — something that will become painfully apparent to most users. See here for more details.
In order of simplicity, these are my recommendations: